On the 80th Birthday of Dmitry Borisovich Fuchs

On the 80th Birthday of Dmitry Borisovich Fuchs††This paper is a contribution
to the Special Issue on Algebra, Topology, and Dynamics in Interaction in
honor of Dmitry Fuchs. The full collection is available at
https://www.emis.de/journals/SIGMA/Fuchs.html

Alice FIALOWSKI ${}^{{\dagger}^{1}}$, Ekaterina FUCHS ${}^{{\dagger}^{2}}$,
Elena FUCHS ${}^{{\dagger}^{3}}$, Boris KHESIN ${}^{{\dagger}^{4}}$,
Alexandre KIRILLOV ${}^{{\dagger}^{5}}$, Fedor MALIKOV ${}^{{\dagger}^{6}}$,
Valentin OVSIENKO ${}^{{\dagger}^{7}}$,
Alexei SOSSINSKY ${}^{{\dagger}^{8}}$ and Serge TABACHNIKOV
${}^{{\dagger}^{9}}$

B. Khesin, F. Malikov, V. Ovsienko, S. Tabachnikov

${}^{{\dagger}^{1}}$ Institute of Mathematics, University of Pécs, Pécs,
Hungary<EMAIL_ADDRESS>http://web.cs.elte.hu/~fialowsk/

${}^{{\dagger}^{2}}$ Department of Mathematics, City College of San Francisco,
CA 94112, USA<EMAIL_ADDRESS>https://www.ccsf.edu/Info/Faculty_In_Review/7272/

${}^{{\dagger}^{3}}$ Department of Mathematics, UC Davis, One Shields Ave
Davis, CA 95616, USA<EMAIL_ADDRESS>https://www.math.ucdavis.edu/~efuchs/

${}^{{\dagger}^{4}}$ Department of Mathematics, University of Toronto,
Toronto, ON M5S 2E4, Canada<EMAIL_ADDRESS>http://www.math.toronto.edu/khesin/

${}^{{\dagger}^{5}}$ Department of Mathematics, University of Pennsylvania,
Philadelphia, PA 19104-6395, USA<EMAIL_ADDRESS>https://www.math.upenn.edu/~kirillov/

${}^{{\dagger}^{6}}$ Department of Mathematics, University of Southern
California, Los Angeles, CA 90089, USA<EMAIL_ADDRESS>https://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1003489

${}^{{\dagger}^{7}}$ CNRS, Laboratoire de Mathématiques de Reims, 51687 Reims
cedex 2, France<EMAIL_ADDRESS>http://ovsienko.perso.math.cnrs.fr

${}^{{\dagger}^{8}}$ Independent University of Moscow, Bolshoy Vlasyevskiy
Pereulok 11, 119002, Moscow, Russia<EMAIL_ADDRESS>https://users.mccme.ru/abs/

${}^{{\dagger}^{9}}$ Department of Mathematics, Pennsylvania State University,
University Park, PA 16802, USA<EMAIL_ADDRESS>http://www.personal.psu.edu/sot2/

Received February 29, 2020; Published online April 07, 2020

This article is a collection of several memories for a special issue of SIGMA
devoted to Dmitry Borisovich Fuchs.

cold topologist; bookcase; new chronology; home-schooling; poetry; mountain
tourism; Lie algebra cohomology; SQuaREs

01A60; 01A61

##  1 Preface by B. Khesin, F. Malikov, V. Ovsienko
and S. Tabachnikov

This special issue of SIGMA is devoted to Dmitry Borisovich Fuchs, an
outstanding mathematician and our teacher, on the occasion of his 80th
birthday. It comprises many papers of his friends and colleagues, reflecting
broad mathematical interests of D.B. himself.

Below we collected several memories: those of his daughters Katia and Lyalya,
of his old-time friends A. Sossinsky and A. Kirillov, and his students and
colleagues A. Fialowski and V. Ovsienko.

There are many (semi-legendary) stories about Fuchs and, by way of
introduction, we present a few specimen of this lore.

Figure 1: Fuchs is an avid bicyclist, making very long distance trips, in
particular, from Moscow to Kazan (about 500 mi). This picture was drawn by
Sergei Ivanov, the artist who illustrated the book “Mathematical Omnibus” by
D. Fuchs and S. Tabachnikov, AMS, Providence, RI, 2007.

In his memories of Arnold’s seminar in Moscow D.B. Fuchs writes (see [2]) “My
role there was well established: I had to resolve any topology-related
difficulty. Some of my friends said that at Arnold’s seminar I was a ‘cold
topologist’. Certainly, a non-Russian speaker cannot understand this, so let
me explain. In many Russian cities there were ‘cold shoemakers’ in the streets
who could provide an urgent repair to your footwear. They sat in their booths,
usually with no heating (this is why they were ‘cold’), and shouted, ‘Heels! …
Soles! …’ So I appeared as if sitting in a cold booth and yelling, ‘Cohomology
rings! … Homotopy groups! … Characteristic classes! …’ ”

The topological reputation of Fuchs extended far beyond Arnold’s seminar.
There was a famous story where a group of mathematicians, “all the best
people” as Fuchs says, helped one of them, A.A. Kirillov, to move to his new
apartment on the 14th floor of a 16-story building. It turned out that a huge
custom-made bookcase, which would occupy the entire wall of a room, did not
fit into the elevator. It had to be carried to the 14th floor via the
staircase but, even more importantly, it was so large that there was a unique
way of moving it through each flight of stairs, with the only possible turn of
the bookcase fitting the stairwell. (One of the participants was Ya.G. Sinai,
who first formulated the problem as “keep this damned bookcase connected” and
later “keep the number of its components not greater than two”.) Eventually,
when they got to the apartment, there was a unique way to bring this bookcase
into the room, with no way to turn it around or change its orientation in any
other way.

And once inside, it turned out that the bookcase fit the room in the only
possible position, with the shelves facing the wall and the backside facing
the room! The only way to fix this was to carry the case all 14 flights down
the stairs and get it out to the street. Then Fuchs, as the leading
topologist, was given the task of finding the correct “initial conditions” in
order to start the process anew. He thought about it, gave the instructions
how to turn it, they did all the lifting via 14 flights of stairs up – and the
bookcase indeed fit in the room perfectly!

S. Tabachnikov recalls (see [4]) “At our first meeting DB asked me what I knew
in mathematics, and I started to name various books I had read (after I
mentioned Spanier’s “Algebraic topology” DB remarked, characteristically, that
it was the first time he met someone who had managed to finish the book). DB
asked a few questions (e.g., which simply connected 4-dimensional manifold I
knew), and my answers revealed the obvious: my extensive reading was almost a
total waste of time, I did not really understand the material. This was a
devastating discovery for me, and I mumbled something like: “I should probably
quit mathematics, I’ve already lost too much time!” This made DB laugh. “You
are 20, aren’t you? It is not too late to start, even if you had never read
anything but ‘Murzilka’111A Russian magazine for preschoolers. before”.

F. Malikov remembers: Fuchs was giving a course on linear algebra and once,
trying to entertain us, he devoted a lecture to a proof of Abel’s theorem
(impossibility of solving equations of degree five in radicals). While proving
it, he constructed a certain permutation. Then unexpectedly he turned to A.
Kanel’, who was the indisputable genius of the class, and asked: “Alesha, what
could you say about this permutation?” Kanel’ replied: “This permutation is
even”. Fuchs said: “Correct, but it’s not its main property – it is the
trivial permutation”.

Figure 2: In the office.

Yet another memory: Fuchs was giving a talk at USC. At some point, he said
that, for a sufficiently small epsilon, such and such held. One of the
attendees, who was drowsing, suddenly woke up and asked how small this epsilon
must be. Fuchs replied immediately: “one tenth of an inch”.

B. Khesin remembers the following story. Fuchs recalled that in the 70–80s he
was at one of A. Fomenko’s public lectures on “The New Chronology”, which was
a “hot” subject at the time. At some point, when comparing time patterns
related to two sequences of the Roman emperors, Fomenko said that the
probability that such patterns were independent and not copied from one
another was one out of a million. At this moment Fomenko tried to clarify his
point: “In order to see how negligible this probability of one millionth is,
just imagine that you put a kettle on a hot stove, but instead of boiling the
water freezes up! This is how rare such events are!” Fuchs said that once he
heard that explanation, he thought: “Hmm … Moscow’s population is 8 million …
So every morning 8 people in this city put their kettles on the stove and have
the water in the kettle frozen?!”

##  2 Elena Fuchs

El. Fuchs

One of my first memories of my father was waking up in the morning to find a
sheet of math problems, handwritten by him and often involving me and my
sister, in my room. My father would be gone to work by this point, but it was
like a little piece of him that he left for me to enjoy.

Figure 3: A page from the problem book that Fuchs wrote for his daughters.

When I was a little older, after we moved to America, these sheets of problems
came illustrated with a sketch of me doing something from one of the problems
(I remember one picture in particular where I was diving into a swimming pool,
something that he had taught me to do, in fact). Not everyone knows that my
father is a wonderful artist: our apartment in Moscow had several of his oil
paintings hanging on the walls. He doesn’t paint much anymore, or talk about
it, but I think he painted those paintings at a sad time in his life when the
art helped him escape, so perhaps that is a good thing.

In fact, I do not remember a time in my life when math did not play a big role
in our relationship. While we still lived in the USSR, I remember that he was
often away on what I imagined to be an exciting adventure off in some other
country. He would always come home with all sorts of treasures from the West
that were unattainable at home at the time: colorful skirts, stuffed animals
that looked like the real thing, fancy new pencils.

Figure 4: The Fuchs family.

While my sister and I and the other kids in our neighborhood lounged in the
sun on a mattress left on a ping-pong table at our dacha (summer home), my
father would have frequent math visitors, and sometimes he would take me from
there into town on his bike so that I could have ice cream while he met with
yet another mathematician. Even when my sister and I did not fully appreciate
it, our lives were engulfed in math. Wherever my father took us: to the
skating rink, to a lesson of some kind, he always had his yellow pad of paper
on which he would write and draw what back then looked like amusing doodles.
During a month-long stay at St. John’s College in Cambridge in the mid-
nineties, we would find his math toys at our apartment when my parents were
away and play ridiculous games with them: one time we found a curvy ruler for
tracing curves that one could shape into any curve one wanted and made it into
an S-shape to advertise Safeway, the American grocery store, out of our
apartment window. I am sure the Brits were amused.

Rooms with floors covered in chalk dust have been places in which I have felt
at home from a very young age, and it is all due to him. Indeed, the first
time I remember anyone asking me what I wanted to be when I grew up,
“mathematician” was always on the long list of professions I planned to have
(along with opera singer, ballerina, and flautist … those did not pan out).

In the US, I acquired most of my mathematical education until I attended
university classes from my parents. There was a designated chair in which my
father would sit by my desk to explain new interesting math to me, and I was
officially home-schooled in math for 2 years by him: I might add that those
two years probably marked the peak of my teenage rebellion, but he patiently
stuck it out, and one lesson I have held on to since those times was to listen
to someone explaining some mathematics to me no matter if I thought I already
knew it: chances are, my father said, I might learn something new. He was
right then and is still right now: I never dismiss mathematics as
uninteresting and always try to learn something from it.

One might think that our close mathematical relationship through my childhood
would mean a continued close mathematical relationship in adulthood. That has
not been true: in college I contemplated majoring in a scientific field
besides math and leaning into it, out of a worry that mathematicians might
think any of my success was due to my father’s support. I remember talking
about things along these lines to my father then, and he would often tell me,
“your problem is not your mathematical ability. It’s your insecurity in it.”
It took me a few semesters in university to see that he was right, and I found
a passion in number theory, which was fairly far removed from his mathematics
(although he has told me at least 50 times that, while he would like to say he
has never worked in number theory, that would not be entirely true). Since
then our mathematical relationship has been much less intense, although I do
still have a dream to produce a Fuchs–Fuchs paper sometime.

I have to add that, part of my father’s dedication to my education in math
certainly came out of his genuine care for teaching: even when he was teaching
students in the university and not his children, he would go above and beyond
the norm, writing his own lecture notes for nearly every class, shaping his
own point of view in a way that I think is rare among university instructors.
But part of it is a reflection of how much he cares for his family. His
passion, I think, is split between two things: family and mathematics. For
every fond memory that I have of us doing math together, I have one where math
was not at all a factor. Putting up a tent together at countless campsites,
hiking dozens of trails together, confiding in him both my biggest joys and
fears. Most recently, I have seen the joy he gets out of seeing his
grandchildren – my two kids – whenever they are around. I feel truly lucky to
have him as my father, and wish him the most wonderful of birthdays, and many
more to come.

##  3 Ekaterina Fuchs

Ek. Fuchs

It seems that the Russian way of raising a baby when I happened to be one
involved a lot of walking, baby in stroller, outdoors. And so we walked, my
Dad and I for hours on end. I would do what babies do, I suppose, and sleep,
while he pushed the stroller, and did what he continues to do constantly –
think and read. He would tell me how when it rained he would put a book into a
clear plastic bag so he could read it without the book getting wet, and we
would walk on, day in and day out.

And then night would fall, and he would read again, or rather recite, poetry
to me until I was soundly asleep. My mom has often told me stories of sitting
in the kitchen (how vividly I remember the layout of our 3 room apartment on
Trinadtsataya Parkovaya, “Thirteenth Park St” in Moscow!) and knowing exactly
how long it would be until he would be finished, since he recited the same
poems every time, in his slow, peaceful baritone. Even when my sister and I
were older, he would often sit with us as we drifted off into sleep, reciting
poetry to us.

Poetry continues to be a key motif in my memories of growing up and my dad.
One summer, if I had to I would guess the summer of 1988, but I can’t be
terribly sure, my father and I decided together that he would help me memorize
the entire anthology of Boris Pasternak poetry that followed the novel “Doctor
Zhivago”. Some days we would work on one poem, sometimes more than one, some
poems stuck almost like honey in my mind, some were very difficult to
internalize. The poems dealt with some very adult themes, and I remember in
many cases our work to memorize the poems had to include long conversations
about what the poem meant, and what the imagery was trying to evoke. We made
it through that entire anthology that summer, just like we planned. To this
day we occasionally allow ourselves a stroll down memory lane, and see if we
can dust off this that or the other poem - while they are rusty and sleepy in
my head, they are unbelievably crisp and alive in his. So many times I’ve said
“I’m going to relearn this one!” and he’s immediately ready to help me with
any word or line where I stumble.

Figure 5: On a hike with Katia and Lyalya.

I suppose this reminds me of another motif, if one could call it that, in my
life with dad. The time he was willing to devote to basically anything we did
together. One year he decided that the whole family needed a good solid lesson
on the history of the Roman Empire. He immersed himself in historical texts
and made copious notes, which he would then use to teach us (this included
myself, my sister, my mom, and the cat). These were incredibly cozy evenings,
sitting on the big L shaped sectional in the dining room, the three of us (and
the cat) huddled together on the long side of the L, him sitting on the short
end. I must confess that the evenings got so cozy, that my sister and the cat
and I would often doze peacefully, which didn’t go over particularly well with
our parents.

He was then and continues to be now, unquenchably thirsty to teach us things –
whether it is beautiful mathematics, or the history of Ancient Rome, or
poetry.

As a teacher now, I know that my desire to help others learn about beautiful
things came from the true great fortune of having been surrounded by
incredible teachers my whole life. My first teacher, without a doubt, was my
dad.

I can honestly say that my dad’s influence can be found at the very most
foundational core of who I am today. His sense of humour, his dedication to
his family, his love of art and thirst for knowledge, have shaped me in
incredibly important ways. Basically, he’s awesome, and I wish him a happy
birthday, and many, many, many more still to come.

##  4 Alexei Sossinsky

A. Sossinsky

I have known Dmitry Fuchs since 1957, when we were both students at the
mathematics department of Moscow State University. We became neighbors quite
by accident (our parents had acquired apartments in adjacent houses in
Izmailovo) in our graduate student days and soon became close friends. This
friendship has remained, despite the fact that, since 1990, we are usually
separated by an ocean and a couple of continents.

Mitya, as everyone calls him is, above all, a mathematician, and while
mathematics dominates his life, he has many other interests and activities
that shape his unique personality: his sense of humor, his fascination for the
outdoors, his love of poetry, his interest in the popularization and teaching
of mathematics (a rare trait for a research mathematician of his caliber). It
is mostly on these not very mathematical aspects of Dmitry Fuchs’ life that I
intend to concentrate here, in the form of “little Fuchs stories” drawn from
my vast oral repertoire.

Figure 6: D. Fuchs, A. Sossinsky, and Katia Fuchs.

Sense of humour. Mitya’s is always low key, often ironic, it permeates his
mathematical lectures and his conversation about people. Here is a related
little story. We – a group mainly consisting of mathematicians – were downhill
skying on the Chiget mountain in the Caucasus. Yulia Gippenreiter, the non-
mathematician in our group, was a psychologist and an accomplished
(professional-level) Alpine skier. She would generously give us tips on skiing
techniques; to Mitya, speeding fearlessly downhill, legs spread far apart for
better balance, she would shout: “Mitya, parallel skis, keep your skis
parallel!” Later, over tea in the cozy Ai chalet, Mitya a would grumble,
“These stupid people in the humanities – they think that if your skis are more
than a meter apart, they can’t possibly be parallel”, and we would all burst
out laughing.

Poetry. We first discussed poetry with Mitya as neighbors, in Izmailovo, and I
must admit that at first – with a youthful snobishness of which I am now
ashamed – I looked down on his knowledge and taste in Russian poetry. The
first “poetry reading” where I began to change my mind occurred in the Kolsky
peninsula in the late sixties, and I have described it in detail elsewhere,
[4], but it still often comes to mind: the three of us (the third was our
friend and colleague Lenya Erdman) on the Lovozero plateau, standing in a
snowstorm near our tent, which was being progressively buried in snow, waiting
for the blizzard to abate and the late Arctic morning to come, and – yes, you
have guessed correctly – reading poetry to each other, sometimes in two or
even three voices. I remember that Mitya got things started by reading a
beautiful poem about Ulysses by Lugovskoi (whom until then I had regarded as a
worthless Soviet establishment poet), Lenya recited a lot of funny poems by
Sapgir and Zakhoder, I remember reciting Lermontov’s Demon and, when my
Russian repertoire had dried up, Poe’s Raven, with my friends pitching in,
like a Greek chorus, shouting out the mandatory “Nevermore” at the end of each
stanza.

A less dramatic, but just as rewarding, reading took place a couple of years
ago at the IHES in Bures-sur-Yvette, where we lived in neighboring cottages
(not by accident, but by design). My present for Mitya’s 80-th birthday was a
selection of poems by Georgy Ivanov, a Russian émigré poet he was not familiar
with, and he responded with a short letter with a very precise literary
analysis of Ivanov’s verse, as good as those ever written by “people in the
humanities”.

Figure 7: Fuchs and Sossinsky follow the donkeys.

The outdoors. Mitya could satisfy his love of nature only within the limits of
the “iron curtain” surrounding the USSR, camping along its rivers, in its
forests and mountains. This he would do within the framework of what we call
sportivny tourism: whitewater trips (shooting rapids in kayaks), mountain
hiking (on foot in summer or on skis in winter), hiking-camping trips in the
rich forests of Russian midland. Of the several camping trips that we took
together, I will describe the only one during which I took photographs: a high
mountain hiking-camping trip through the Fanskie mountains, from Dushanbe to
Samarkand. It began from the northern outskirts of Dushanbe, from which we
began the long climb up with our backpacks loaded on two donkeys (see photo),
until the path became so steep that the poor overloaded donkeys refused to
continue. Progressively getting used to the increasing altitude, we hauled
ourselves and our backpacks up to the first mountain pass, Mura, at 3500
meters.

Figure 8: Fuchs rock climbing.

From then on, it was down and then up over higher and higher mountain passes
(the highest was at 4900 meters), crossing ice-cold streams, overcoming steep
glaziers and rock climbing. Another photograph shows the group resting at the
start of one of the glaciers, with two local shepherds gratefully sharing our
meal. The local population (nomadic shepherds) was very sparse and practically
untouched by 20th century civilization, their self-sufficient way of life
unchanged for many centuries, most of them having never handled money, been in
a city, or seen a photo-camera; they looked at us as if we were creatures from
another planet.

Figure 9: A frugal meal at the foot of a glacier.

One of the high moments of our trip (literally and figuratively) was in the
morning at our camp site some 4000 meters above sea level, when a huge eagle
swooped down on us, grabbed a sausage that Mitya was about to cut up into
seven equal pieces and flew off with it; I think it was then that Mitya
philosophically declared: “Well, the sausage was beginning to spoil anyway”.
The whole episode took two or three seconds, and so, unfortunately, there is
no photograph of the eagle, and there are no photographs that do justice to
the mountain scenery and the beautiful and then almost totally uninhabited
Iskander Kul’ lake (now a tourist resort), on whose shores we rested for a
couple of days before finishing our trip in the historic city of Samarkand.

Figure 10:  A. Sossinsky, I. Girsanov, D. Fuchs, E. Mochigin, G. Turina, Yu.
Chapovsky (left to right).

There is, however, one photograph that is really worth looking at. It very
expressively shows six tired climbers, some still breathing heavily, who have
just overcome a mountain pass and look forward with optimism to the continuing
trip. Yet this is a tragic photo: five years after it was taken, only two of
the six persons on the picture were still in this world: Girsanov, Mochigin,
and Chapovsky were buried in an avalanche in the Sayany mountains in 1967 and
Galya Turina, Mitya’s wife, drowned two summers later when her kayak
overturned in a white water trip in 1970 just above the Arctic Circle in the
Urals. But Mitya and I are still here and look back on this trip as one of the
most rewarding experiences of our lives.

* * *

Perhaps I should conclude this text by shortly recalling how our life lines
converged and diverged in the last 60 years. From our graduate years on, our
lives first developed in parallel, with Mitya one year ahead of me (although
almost two years younger, he was a year ahead in his studies). After
completing our graduate studies and defending our PhD’s, we became, one after
the other, associate professors at the topology Chair of P.S. Alexandrov
(although neither of us were his pupils, and we worked in the rapidly
developing field of algebraic topology that Alexandrov no longer understood).
We both became regular participants in the famous Gelfand seminar, and also
worked together in the math olympiad movement on the national and
international level.

I recall an episode that occurred at the Gelfand seminar: during a break, I
walked up to the blackboard and tried to explain to Fuchs how to answer a
question raised at the talk preceding the break; this involved the calculation
of the first term of a spectral sequence, I explained the filtration and, with
Mitya’s help, indicated how the calculation would go. I did not notice that
Gefand was standing by and attentively listening; when I had finished and the
seminar was about to resume, Izrael Moiseevich, addressing us both, said that
was quite interesting and asked us to write it up and bring the text next
week, he would have it published as a joint paper by the three of us. But we
didn’t: Mitya convinced me that the question was not very serious and the
proof too simple to be worth publishing.

Perhaps a year later, at the end of the seminar, Gelfand took us aside and
proposed us research positions at his mathematical biology laboratory. In my
case, I think it was the episode described above which decided that I was
worth hiring; as to Mitya, it was by then clear that he was a first rate
mathematician who would embellish any research center. Mitya accepted. I
refused: I was simply afraid of Gelfand.

This was a second step in our moving apart, the first one being geographical:
I moved to an apartment in the southern part of town, so we were no longer
neighbors. The next step in that direction occurred when I was forced out of
Moscow University, and after a year of unemployment, started working at the
popular science magazine Kvant. During the 13-year period that I worked there,
we would meet with Mitya at the Kvant headquarters more often than at research
seminars. Mitya became a regular contributor to the magazine, and an
unexpected result of that was his marriage to Ira Klumova, who then worked as
a Kvant math editor. It was my wife, Elena Efimova, not I, who played the role
of the matchmaker: two side-by-side tickets to a theater performance given by
Lena to Mitya and Ira started things off …

Another place where we continued to meet was Bella Subotovskaya’s People’s
University (while it lasted), where Mitya, in this case following my
footsteps, became an active teacher until this “university” was closed down by
the KGB. I have described the dramatic story of that unusual institution in my
article [3].

One of our last contacts in Moscow before Mitya’s emigration to the US in 1990
was a series of talks on knot polynomials that I gave at the Fuchs–Varchenko
seminar. This was extremely important for me – Mitya’s very favorable reaction
and encouragement helped me realize that during my “exile” to Kvant I did not
lose the ability of doing serious mathematics.

Since 1990, our meetings have been few and far between: at Davis, of course,
at Oberwolfach, at the IHES, in Moscow at the Gelfand Centenary celebration –
are those that first come to mind.

Birthday wishes? I have sent them to Mitya by e-mail: to be able to think
clearly and mathematically till the very end, to derive from life whatever
pleasures we are still capable of appreciating, and, our healths and God
willing – a few more meetings on either side of the ocean.

##  5 About Mitya Fuchs. By Alexandre Kirillov

A. Kirillov

I was asked to write something for the special issue in honor of 80th
anniversary of D.B. Fuchs, my old friend, an outstanding mathematician, and
simply a very good man. Of course, I immediately agreed. But when push came to
shove, it turned out that to do it wasn’t that easy. In general, it is not
easy to write a good mathematical paper, and to write it better (or at least
not worse) than one’s previous works is even harder. Let alone on a short
notice …

Of course, one could simply write about Mitya, how we lived, worked, what we
did in our free time. But this immediately forces one to remember many other
people and all our life in the USSR, Russia, and America because in all of
this Mytya played an important role. And still, I shall give it a try.

Figure 11: Left: as part of Khrushchev’s campaign to open vast tracts of
virgin land in the northern Kazakhstan and the Altai region of the USSR, the
students were sent to work there in summer. Right: a young topologist, D.
Fuchs.

Much was said about the Golden Years of Moscow mathematics (1955–1967). We
both had the privilege to study at MekhMat of MGU (Mathematics Department of
the Moscow State University) that, without exaggeration, was then the leading
mathematical center in the world.

Among our professors were P.S. Alexandrov, N.V. Efimov, I.M. Gelfand, A.O.
Gelfond, B.N. Delone, E.B. Dynkin, A.N. Kolmogorov, A.G. Kurosh, A.A. Markov,
D.E. Menshov, I.G. Petrovski, L.S. Pontryagin, M.M. Postnilov, I.R.
Shafarevich, B.V. Shabat, G.E. Shilov. The recitations and research seminars
for youngsters were run by then young, but already well known, mathematicians
F.A. Berezin, R.L. Dobrusin, R.A. Minlos, A.S. Schwarz, A.G. Vitushkin.

The general atmosphere of those years (1955–1967) was splendid, and it has
never been the same again. What happens at MekhMat today does not stand
comparison. I was lucky to belong to a group of 20–25 students who were united
in their admiration for talent and were genuinely interested in each other’s
work. (To mention only the most active and well remembered: A. Arkhangelskij,
V. Arnold, B. Averbuch, N. Brushlinskaya, A. Chernavskij, L. Churaeva, V.
Murskij, R. Sofronitskaya, M. Shur, V. Tutubalin, F. Vetukhnovskij, E.
Vinberg.)

Later our group extended by students from the next year (D. Fuchs, V.
Palamodov, A. Sossinsky, G. Tyrina) and some previous years (V. Alexeev, I.
Girsanov, B. Polyak, Ya. Sinai, V. Tikhomirov).

Besides mathematics, the common trend was tourism. Not in the western form of
expensive tours in comfortable conditions, but week-end pedestrian hiking on
foot near Moscow, or longer travels during student vacations on ski, on
kayaks, in the mountains. These travels were partially sponsored by the
University Tourist Club. One can get some outfit (skis, kayaks, ropes) and
some foodstuffs (condensed milk, braised meat) which was rather difficult to
find in the post-war Russia.

Mitya was a very important member of our expeditions due to his strength,
endurance, and good nature. Once he saved our kayak when we tried to pass in
poor visibility through a narrow passage in a dam. The kayak was almost folded
in half, being pressed by the waterfall and only Mitya’s power legs and spine
allow us to pull it out.

Another remarkable situation happened during our winter expedition on the
Northern Ural, the coldest place in European Russia, where the temperature in
winter falls to $-50$ C∘. By the way, one of our common discovery with Mitya
was that, despite the general opinion, the spittle (and other warm excretions)
do not freeze immediately in the air even under this temperature.

Preparing to this expedition, we decided to make an experiment and take with
us a portable stove working on firewood (the gas stoves did not yet exist in
tourist practice at that time). On the third or fourth day, when the frost
became rather severe, we installed the stove in our tent and started the fire.
Soon the air inside the tent became rather warm, but the whole space filled up
by the dense smoke. Since none of us had experience of heating a tent with a
stove, we thought that it was a natural effect and tried to endure the smoke.
But after a short time the smoke turned into flame, and when we finally
succeeded to put it out, almost half of our tent had gone.

The culprit of the fire was the veneer shovel, incautiously leaned against the
stove. (Such shovels are usually used in winter expedition for digging the
snow to make a place for the tent and for the campfire.) In the morning there
was a council about one main question: what to do? Some people proposed to fix
the tent and to go further. Other preferred to break the expedition and come
back home.

Since I was the official head of expedition, my vote was decisive. And with
great doubt I decided to return. I was very thankful to Mitya and his wife
Galya who did not blame me for this decision, though I suspect they felt
differently. To my surprise, the way back was rather easy and agreeable.
First, going down the river and using our trace in snow, we made the way back
in one day (more precisely, in 12 hours). Second, the frost was softened and
when we came to the railroad station, the temperature was $-18$ C∘, feeling
like a summer day.

The mathematical achievements of Mitya are numerous and well-known. Last
several years we tried to work together. Luckily, I draw to this project my
former PhD student V. Ovsienko and his wife S. Morier-Genoud. Thanks to
Valya’s administrative talent and modern social trends in scientific politics,
the four of us formed a “Square” team at the American Institute of Mathematics
(Palo Alto – San Jose).

We spend at AIM three one-week terms, and wrote the joint article, where we
tried to realize my idea about the application of the orbit method to the
group of triangular matrices over a finite field. I’ll explain here the
problem in very informal way.

The point is that usually the linearized version of a problem is simpler than
the non-linear original. But the classification problem for coadjoint orbits
of the triangular matrix group $N$ is still open, notwithstanding that it is
the linearized version of the classification problem for the Borel group
orbits in the flag manifold $F$. The solution of the latter problem is well-
known: the B-orbits are labeled by the elements of the Weyl group. I
conjectured that the coadjoint orbits are in some sense the “shadows” of
$B$-orbits. The naive formalization of this conjecture turns out to be wrong,
but we have found a very interested new phenomena and are eager to continue.

I can not speak about Mitya and not to mention his family. His first wife,
Galina Tyurina, was an ardent tourist and a downhill skier. She was also an
outstanding mathematician and an essential member of I.R. Shafarevich’s team
in algebraic geometry. She died tragically during a kayaking expedition.

The second Mitya’s wife, Irina Klumova, was my student at MechMat and later my
co-author in the magazine “Kvant”, where she worked for many years. They have
two beautiful daughters, Katia and Elena, both talented mathematicians. Thanks
to Elena, Mitya became a happy grandfather of two.

Figure 12: D. Fuchs, A. Kirillov, and I. Klumova. Fuchs, the grandfather.

Now we meet not as often as we wish, but I still look forward in the hope on
the further joint work.

##  6 Close friendship with the Fuchs family in Moscow.
By Alice Fialowski

A. Fialowski

When I was a graduate student (an aspirant) in Moscow from 1979 to 1983, I
lived in the dormitory of the Moscow State University, MGU. I regularly
visited the Gelfand Seminar on Monday evenings. In a sense, it was a big
family in the lecture room: students and professors were talking to each
other, made friends there. One of the main people at those seminars was Dmitry
Fuchs, who usually sat with Alexandre Kirillov in the first row. Gelfand often
pestered them with questions.

My first year as an aspirant was quite hard. Partially it was related to the
fact that I did not have much background in the topics which were discussed.
MGU was the very best school of mathematics in the entire world at the time,
and the research topics there were numerous and at the highest international
level. In comparison, the Hungarian school was mainly influenced by Erdös and
combinatorics in those days. I liked algebra and functional analysis, and
started studying Lie groups, Lie algebras, and representation theory. I was
also learning Lie algebra cohomology from Fuchs, and from his famous book
which was already available in Russian. In order to use this theory for my
project in algebras, I needed help.

Figure 13: The Fuchs family. Ekaterina Ivanovna.

In those days in Moscow professors did not have offices and used to work with
their colleagues at home. Fuchs invited me to his home and I went to visit him
and his mother, Ekaterina Ivanovna. We had dinner together, wonderful Russian
salads and other stuff. I was touched by their openness and hospitality. We
worked for a few hours, decided to meet the next week, then the next, and thus
our Wednesday meetings became regular.

Of course, we discussed many subjects: the Fuchses were interested in my
country, my family, and Dmitry Borisovich explained to me some details about
the situation in Russia, talked about his brother, his father, his family. I
met Ira, his wife, I was there when they had their first baby Katia, and also
when their second baby, Lyalya, was born. I also went with them to their
dacha. I will forever be grateful to them for making me feel a part of their
family, and the Wednesday meetings were really important to me. Living alone
in Moscow in those days as a foreigner was not so easy, but this way I got a
family there!

I became especially close with Ekaterina Ivanovna, Fuchs’s mother. She was a
lovely lady, and there was something warm and reassuringly peaceful about her.
When I was depressed (which unfortunately happened from time to time), she was
always able to help me get away from feeling down. I was also there when she
was dying. She died in her family, in peace.

After my thesis defense, I returned home to Budapest, but I remained in
regular contact with my “Russian family”. Fuchs, Ira, and the girls visited me
in Budapest, and I was able to take them to nice playgrounds on the Gellért-
hegy, and other places. Dmitry Borisovich had many good friends in Budapest,
like Károly Mályusz, András Krámli, Péter Major, Doma Szász, and others. They
all got acquainted with Dmitry Borisovich in Moscow, and his lovely
personality attracted many-many people. (Unfortunately Károly and András are
not with us anymore …)

Figure 14: The Fuchs family.

Soon after I returned home, I began to get invitations to the West. I was able
to go to Germany for a Humboldt fellowship and to attend various conferences.
I also received an invitation to the University of Pennsylvania and later I
got a permanent job at the University of California at Davis. While I was
there I learned that Dmitry Borisovich might be able to come to the United
States with his family. I immediately suggested to our Department Chair to
invite him to join our Faculty. Eventually, it worked out and since then
Dmitry Borisovich Fuchs, together with Albert Schwarz, have graced the Davis
mathematics community. UCDavis is lucky to have them! Recently Fuchs’s younger
daughter, Elena also joined the Faculty, together with her husband, Martin
Thanh Luu.

I was able to continue working with Fuchs in Davis as well. We lived just a
few houses apart, and I often visited the Fuchs family there too. After my
return to Hungary, we have not seen each other very often. But I went to Davis
for his 70th birthday celebration, and we also met every other summer at the
Max-Planck Institute in Bonn, where both his family and mine usually spend one
month.

I feel lucky that I have such a real friend as Dmitry Borisovich. For me he is
not just a colleague with whom it is good to work, but a close friend, with
whom we share a lot of values in life. I wish him all the very best and long
and happy years with his family!

##  7 Personal notes by Valentin Ovsienko

V. Ovsienko

What can be added to the beautiful (and sometimes romantic) descriptions of
the mathematical life in Moscow just before the Perestroika? Many remarkable
memories have been published in recent decades by various authors. “Those days
are past now and in the past they must remain …” I can only talk about
personal impressions of a young undergraduate student that I was at that time.

The real life at Mekhmat started in the afternoon, and I spent them all
running from one fascinating seminar to another. This is how I met Dmitry
Borisovich. I saw him several times every week, at Gelfand’s seminar (that, as
we know, was the main mathematical event), Arnold’s seminar, meetings of the
Moscow Mathematical Society, etc. Fuchs gave several lectures at Math
Society,222A series of survey introductory lectures for a large audience.
where he talked on various subjects. I remember very well his lecture about
symplectic topology, a new domain that was making its “first steps”. It was
not at all (not yet) among the subjects of his own research, but Fuchs gave a
very clear and precise overview. I was not at all surprised, it was obvious to
me that Fuchs knew everything!

My first personal meeting with Dmitry Borisovich was related to the seminar of
Arnold. I was a regular member of the seminar, and Arnold invited me to give a
talk about the “Lagrange Schwarzian derivative”, a notion introduced in one of
my first research papers. That was my first talk outside Kirillov’s seminar
for younger students, and I was very anxious. Being out of town, Arnold asked
Fuchs to run the seminar, and Dmitry Borisovich took the job seriously: he
suggested to meet and discuss a few days before the seminar. Our meeting
became for me one of the episodes that remain in memory for the rest of the
life. Fuchs asked many questions and made several comments, that later I used
many times. Trying to explain my work to him, I got an impression of
understanding it myself. My talk at the seminar went smoothly …

My research was largely influenced by the famous papers authored by
Gelfand–Fuchs and Feigin–Fuchs. With varied success, I tried to connect their
cohomology classes to projective differential geometry, and this task occupied
me for many years. During these long years I met Dmitry Borisovich from time
to time at mathematical conferences and, even more often, at the Tabachnikovs’
home. But never could I imagine that I would collaborate with him, until such
an opportunity suddenly presented itself. This happened 30 years after the
events described above.

Figure 15: Dmitry Borisovich Fuchs.

Five years ago, my teacher, Alexandre Alexandrovich Kirillov, who has always
been a close friend of Dmitry Borisovich, suggested to work on an old unsolved
problem of classification of coadjoint orbits of the nilpotent Lie algebra of
(upper) triangular matrices. Kirillov created a “square-team” consisting of
four people: Fuchs, Kirillov, Sophie Morier-Genoud, and myself. We worked
within the SQuaREs program at AIM, and later, at MFO, Oberwolfach. Let me
mention that during our work on the project that lasted for three years and a
half, the square eventually became a pentagon.

This work with my teachers was yet another moving experience. Fuchs and
Kirillov stayed at the blackboard, intensively discussing the subject and
paying almost no attention to us. Kirillov directed the general attack, Fuchs
generated the technique. Sometimes they asked Sophie about details of the
modern combinatorial theories. I was the chronicler, always with a notebook at
hand, and I recorded each word of the “oracles”.

In the afternoon, the discussion usually became more relaxed. “Do you remember
when we climbed Tian Shan Mountain in 1956?” would ask one of them. “Of
course, I remember”, would reply the other, “but it was in 1957!”

Gradually, our subject evolved. Approximating the orbits, we found ourselves
on a more solid soil, studying Schubert varieties and their tangent cones.
Little by little, some understanding and some results started to appear.

However, writing of our paper [1] was not an easy promenade. The first version
contained two theorems that would solve our problem completely. I was in
charge of reconstructing the details of the proofs, outlined on the blackboard
by the collective efforts, and spent many sleepless nights on that. Finally, I
wrote an SOS message to my coauthors, asking for a permission to call the
second theorem a “conjecture”. When permission was granted, Sophie found a
counterexample to this brand new conjecture. And, finally, the “only if” part
of the first theorem also became a conjecture, but fortunately, with no
counterexamples in sight.

## References

  * [1] Fuchs D., Kirillov A., Morier-Genoud S., Ovsienko V., On tangent cones of Schubert varieties, Arnold Math. J. 3 (2017), arXiv:1606.07846.
  * [2] Khesin B.A., Tabachnikov S.L. (Editors), ARNOLD: Swimming against the tide, Amer. Math. Soc., Providence, RI, 2014.
  * [3] Sossinsky A., In the other direction, in Golden Years of Moscow Mathematics, Hist. Math., Vol. 6, Amer. Math. Soc., Providence, RI, 1993, 223–243.
  * [4] Sossinsky A., Tabachnikov S., Roger C., Astashkevich A., Feigin B., Schwarz A., Personal notes, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 301–313.


# T-Quadratic Forms and Spectral Analysis of T-Symmetric Tensors

Liqun Qi and Xinzhen Zhang Department of Applied Mathematics, The Hong Kong
Polytechnic University, Hung Hom, Kowloon, Hong Kong, China;
(liqun.qi@polyu.edu.hk).School of Mathematics, Tianjin University, Tianjin
300354 China; (xzzhang@tju.edu.cn). This author’s work was supported by NSFC
(Grant No. 11871369).

###### Abstract

An $n\times n\times p$ tensor is called a T-square tensor. It arises from many
applications, such as the image feature extraction problem and the multi-view
clustering problem. We may symmetrize a T-square tensor to a T-symmetric
tensor. For each T-square tensor, we define a T-quadratic form, whose variable
is an $n\times p$ matrix, and whose value is a $p$-dimensional vector. We
define eigentuples and eigenmatrices for T-square tensors. We show that a
T-symmetric tensor has unique largest and smallest eigentuples, and a
T-quadratic form is positive semi-definite (definite) if and only if its
smallest eigentuple is nonnegative (positive). The relation between the eigen-
decomposition of T-symmetric tensors, and the TSVD of general third order
tensors are also studied.

Key words. T-square tensors, T-symmetric tensors, T-quadratic forms,
eigentuples.

AMS subject classifications. 15A69, 15A18

##  1 Introduction

We call a third order tensor ${\mathcal{A}}\in\Re^{n\times n\times p}$ a
T-square tensor. It was called an f-square tensor in [6]. The representation
tensor $\mathcal{Z}\in\Re^{n\times n\times p}$ arising in the multi-view
clustering problem [2] and the multi-view image feature extraction problem [9,
10] is a T-square tensor. Here $n$ is the number of the samples in the
database, $p$ is the number of the views.

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-square tensor.
Let $X\in\Re^{n\times p}$. We may regard $X$ as a tensor
${\mathcal{X}}\in\Re^{n\times 1\times p}$. Define

$F_{\mathcal{A}}(X):={\mathcal{X}}^{\top}*{\mathcal{A}}*{\mathcal{X}},$ (1.1)

where $*$ is the T-product operation introduced in [1, 3, 4], and $\top$ is
the transpose operation in the T-product sense. In the next section, we will
review the definition of T-product and its transpose concept. We call
$F_{\mathcal{A}}$ the T-quadratic form defined by ${\mathcal{A}}$. Then for
any $X\in\Re^{n\times p}$, $F_{\mathcal{A}}(X)\in\Re^{p}$. If
$F_{\mathcal{A}}(X)\geq{\bf 0}$ for any $X\in\Re^{n\times p}$, then we say
that the T-quadratic form $F_{\mathcal{A}}$ is T-positive semi-definite. If
$F_{\mathcal{A}}(X)>{\bf 0}$ for any $X\in\Re^{n\times p}$, then we say that
the T-quadratic form $F_{\mathcal{A}}$ is T-positive definite.

The T-positive semidefiniteness (definiteness) concept here is different from
the T-positive semidefiniteness (definiteness) concept discussed in [12]. The
T-positive semidefiniteness (definiteness) concept in [12] is in the sense of
nonnegative (positive) scalars. Here, the concept is in the sense of
nonnegative (positive) vectors. Thus, the T-positive semidefiniteness
(definiteness) concept here is stronger and may reflect more correlative
properties of T-square tensors.

The T-product operation, TSVD decomposition and tubal ranks were introduced by
Kilmer and her collaborators in [1, 3, 4]. It is now widely used in
engineering [2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In [1], Bradman
defined real eigentuples and eigenmatrices for third order tensors in
$\Re^{n\times n\times n}$. Viewing the wide applications of T-product, TSVD
decomposition and tubal ranks, the theory of eigentuples and eigenmatrices
deserves to be further studied.

In this paper, we extend the concepts of eigentuples and eigenmatrices to
T-square tensors and allow complex eigentuples and eigenmatrices. We show that
an $n\times n\times p$ T-symmetric tensor has unique largest eigentuple ${\bf
s}_{1}\in\Re^{p}$ and unique smallest eigentuple ${\bf s}_{n}\in\Re^{p}$ such
that any real eigentuple ${\bf s}$ of ${\mathcal{A}}$ satisfies ${\bf
s}_{1}\geq{\bf s}\geq{\bf s}_{n}$. We further show that a T-quadratic form is
positive semidefinite (definite) if and only if the smallest eigentuple of the
corresponding T-symmetric tensor is nonnegative (positive).

The T-quadratic function $F_{\mathcal{A}}$ maps $\Re^{n\times p}$ to
$\Re^{p}$. Its positive semidefiniteness (definiteness) requires $p$ quadratic
polynomials of $np$ variables to be nonnegative (positive) simutaneously. We
present its spectral conditions. This theory is noval.

We then further study the relation between the eigen-decomposition of
T-symmetric tensors, and the TSVD of general third order tensors.

The reset of this paper is distributed as follows. We deliver some preliminary
knowledge of T-product operations in the next section. In Section 3, we define
eigentuples and eigenmatrices for a T-square tensor, and show the existence of
the largest and the smallest eigentuples of a T-symmetric tensor. In Section
4, we prove that a T-symmetric tensor is positive semidefinite (definite) if
and only if its smallest eigentuple is nonnegative (positive). We study the
relation between the eigen-decomposition of T-symmetric tensors, and the TSVD
of general third order tensors in Section 5.

##  2 Preliminaries

Let ${\bf a}=(a_{1},a_{2},\cdots,a_{p})^{\top}\in\mathbb{C}^{p}$. Then

${\rm circ}({\bf a}):=\left(\begin{aligned} a_{1}\
&a_{p}&a_{p-1}&\cdots&a_{2}\ \\\ a_{2}\ &a_{1}&a_{p}\ &\cdots&a_{3}\ \\\
\cdot\ \ \ &\ \cdot&\cdot\ \ &\cdots&\cdot\ \ \ \\\ \cdot\ \ \ &\ \cdot&\cdot\
\ &\cdots&\cdot\ \ \ \\\
a_{p}&a_{p-1}&a_{p-2}&\cdots&a_{1}\end{aligned}\right),$

and circ${}^{-1}($circ$({\bf a})):={\bf a}$.

Suppose that ${\bf a},{\bf b}\in\mathbb{C}^{p}$. Define

${\bf a}\odot{\bf b}={\rm circ}({\bf a}){\bf b}.$

In [3], ${\bf a},{\bf b}\in\Re^{p}$ are called tubal scalars. Here, we extend
them to $\mathbb{C}^{p}$.

In general, ${\bf a}\odot{\bf b}\not={\bf b}\odot{\bf a}$. We denote

${\bf a}^{\odot 2}:={\bf a}\odot{\bf a}.$

If ${\bf a}\in\Re^{p}$ is nonnegative, then ${\bf a}^{\odot 2}$ is also
nonnegative. However, if ${\bf b}\in\Re^{p}$ is nonnegative, there may be no
${\bf a}\in\Re^{p}$ such that ${\bf a}^{\odot 2}={\bf b}$. For example, let
$p=2$, ${\bf a}=(a_{1},a_{2})^{\top}$, ${\bf b}=(b_{1},b_{2})^{\top}$ and
${\bf a}^{\odot 2}={\bf b}$. Then we have $b_{1}=a_{1}^{2}+a_{2}^{2}$ and
$b_{2}=2a_{1}a_{2}$. To satisfy these two equations, we must have $b_{1}\geq
b_{2}$. We say that ${\bf b}\in\Re^{p}$ is a square tubal scalar if it is
nonnegative and there is an ${\bf a}\in\Re^{p}$, such that ${\bf a}$ is
nonnegative and ${\bf a}^{\odot 2}={\bf b}$.

For ${\bf a}=(a_{1},\cdots,a_{p})^{\top}\in\Re^{p}$, denote $|{\bf
a}|:=(|a_{1}|,\cdots,|a_{p}|)^{\top}$.

Question 1 Suppose that ${\bf b}\in\Re^{p}$ is a square tubal scalar. Is there
a unique ${\bf a}\in\Re^{p}$, such that ${\bf a}$ is nonnegative and ${\bf
a}^{\odot 2}={\bf b}$?

###### Proposition 2.1

$(\mathbb{C}^{p},+,\odot)$ is a commutative ring with unity ${\bf
e}=(1,0,\cdots,0)^{\top}\in\mathbb{C}^{p}$, where $+$ is the vector addition.

Proposition 2.1 extends Theorem 3.2 of [1] from $\Re^{p}$ to $\mathbb{C}^{p}$,
as we need to consider complex eigentuples for third order real tensors. The
proof is almost the same. Hence, we omit the proof.

Note that the operation $\odot$ is different from vector convolution. For
${\bf a},{\bf b}\in\mathbb{C}^{p}$, the vector convolution of ${\bf a}$ and
${\bf b}$ is in $\mathbb{C}^{2p-1}$.

For $X\in\mathbb{C}^{n\times p}$ and ${\bf a}\in\mathbb{C}^{p}$, define

${\bf a}\circ X=X{\rm circ}({\bf a}).$

###### Proposition 2.2

Let ${\bf a},{\bf b}\in\mathbb{C}^{p}$, and $X,Y\in\mathbb{C}^{n\times p}$.
Then

1\. ${\bf a}\circ(X+Y)={\bf a}\circ X+{\bf a}\circ Y$;

2\. $({\bf a}+{\bf b})\circ X={\bf a}\circ X+{\bf b}\circ X$;

3\. ${\bf a}\circ({\bf b}\circ X)=({\bf a}\odot{\bf b})\circ X$;

4\. Let ${\bf e}=(1,0,\cdots,0)^{\top}\in\mathbb{C}^{p}$ as in Proposition
2.1. Then ${\bf e}\circ X=X$ for all $X\in\mathbb{C}^{n\times p}$.
Furthermore, ${\bf e}$ is the unique element in $\mathbb{C}^{p}$ with this
property.

Proof This proposition extends Theorem 3.5 of [1] from $\mathbb{C}^{p\times
p}$ to $\mathbb{C}^{n\times p}$, except the second half of item 4 is
additional. The proof of the other part except the second half of item 4 is
almost the same as the proof of Theorem 3.5 of [1]. We omit this part and now
prove the second half of item 4. Suppose ${\bf a}\circ X=X$ for all
$X\in\mathbb{C}^{n\times p}$. Then $X{\rm circ}({\bf a})=X$ for all
$X\in\mathbb{C}^{n\times p}$. This implies ${\rm circ}({\bf a})=I_{p}$, the
identity matrix of $\Re^{p\times p}$. Thus, ${\bf a}={\rm
circ}^{-1}(I_{p})={\bf e}$. . $\Box$

For a third order tensor ${\mathcal{A}}\in\Re^{m\times n\times p}$, its
frontal slices are denoted as $A^{(1)},\cdots,A^{(p)}\in\Re^{m\times n}$. As
in [1, 3, 4], define

${\rm bcirc}({\mathcal{A}}):=\left(\begin{aligned} A^{(1)}\
&A^{(p)}&A^{(p-1)}&\cdots&A^{(2)}\ \\\
A^{(2)}&A^{(1)}&A^{(p)}&\cdots&A^{(3)}\\\ \cdot\ \ \ &\ \cdot&\cdot\ \
&\cdots&\cdot\ \ \ \\\ \cdot\ \ \ &\ \cdot&\cdot\ \ &\cdots&\cdot\ \ \ \\\
A^{(p)}&A^{(p-1)}&A^{(p-2)}&\cdots&A^{(1)}\end{aligned}\right),$

and bcirc${}^{-1}($bcirc$({\mathcal{A}})):={\mathcal{A}}$.

Various T-product structured properties of third order tensors are based upon
their block circulant matrix versions. For a third order tensor
${\mathcal{A}}\in\Re^{m\times n\times p}$, its transpose can be defined as

${\mathcal{A}}^{\top}={\rm bcirc}^{-1}[({\rm birc}({\mathcal{A}}))^{\top}].$

This will be the same as the definition in [1, 3, 4]. The identity tensor
$\mathcal{I}_{nnp}$ may also be defined as

$\mathcal{I}_{nnp}={\rm bcirc}^{-1}(I_{np}),$

where $I_{np}$ is the identity matrix in $\Re^{np\times np}$.

However, a third order tensor $\mathcal{S}$ in $\Re^{m\times n\times p}$ is
f-diagonal in the sense of [1, 3, 4] if all of its frontal slices
$S^{(1)},\cdots,S^{(p)}$ are diagonal. In this case, bcirc$(\mathcal{S})$ may
not be diagonal.

For a third order tensor ${\mathcal{A}}\in\Re^{m\times n\times p}$, it is
defined [1, 4] that

${\rm unfold}({\mathcal{A}}):=\left(\begin{aligned} A^{(1)}\\\ A^{(2)}\\\
\cdot\ \ \\\ \cdot\ \ \\\ \cdot\ \ \\\
A^{(p)}\end{aligned}\right)\in\Re^{mp\times n},$

and fold$($unfold$({\mathcal{A}})):={\mathcal{A}}$. For
${\mathcal{A}}\in\Re^{m\times s\times p}$ and $\mathcal{B}\in\Re^{s\times
n\times p}$, the T-product of ${\mathcal{A}}$ and $\mathcal{B}$ is defined as
${\mathcal{A}}*\mathcal{B}:=$
fold$($bcirc$({\mathcal{A}})$unfold$(\mathcal{B})\in\Re^{m\times n\times p}$.
Then, we see that

${\mathcal{A}}*\mathcal{B}={\rm bcirc}^{-1}({\rm bcirc}({\mathcal{A}}){\rm
bcirc}(\mathcal{B})).$

Thus, the bcirc and bcirc-1 operations not only form a one-to-one relationship
between third order tensors and block circulant matrices, but their product
operation is reserved.

The Standard Form of a Real f-Diagonal Tensor. Let
$\mathcal{S}=(s_{ijk})\in\Re^{m\times n\times p}$ be a f-diaginal tensor. Let
${\bf s}_{j}=(s_{jj1},{\bf s}_{jj2},\cdots,s_{jjp})^{\top}$ be the $jj$th tube
of $\mathcal{S}$ for $j=1,\cdots,\min\\{m,n\\}$. We say that $\mathcal{S}$ is
in its standard form if ${\bf s}_{1}\geq{\bf s}_{2}\geq\cdots\geq{\bf
s}_{\min\\{m,n\\}}$.

##  3 Eigentuples and Eigenmatrices of T-Square Tensors

For a matrix $X\in\mathbb{C}^{n\times p}$, let its column vectors be ${\bf
x}^{(1)},\cdots,{\bf x}^{(p)}$. Define

${\rm unfold}(X):=\left(\begin{aligned} {\bf x}^{(1)}\\\ {\bf x}^{(2)}\\\
\cdot\ \ \\\ \cdot\ \ \\\ \cdot\ \ \\\ {\bf
x}^{(p)}\end{aligned}\right)\in\mathbb{C}^{np},$

and fold$($unfold$(X)):=X$. Then we define the T-product of ${\mathcal{A}}$
and $X$ as

${\mathcal{A}}*X={\rm fold}({\rm bcirc}({\mathcal{A}}){\rm unfold}(X)).$

Thus, ${\mathcal{A}}*X\in\mathbb{C}^{m\times p}$.

We now define eigentuples and eigenmatrices of T-square tensors. Suppose that
${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-square tensor,
$X\in\mathbb{C}^{n\times p}$ and $X\not=O$, ${\bf d}\in\mathbb{C}^{p}$, and

${\mathcal{A}}*X={\bf d}\circ X.$ (3.2)

Then we call ${\bf d}$ an eigentuple of ${\mathcal{A}}$, and $X$ an
eigenmatrix of ${\mathcal{A}}$, corresponding to the eigentuple ${\bf d}$.

The eigentuple and eigenmatrix concepts extend the eigentuple and eigenmatrix
concepts of [1] from $\Re^{p\times p\times p}$ to $\Re^{n\times n\times p}$
and allow complex eigentuples and eigenmatrices.

We aim to study T-positive semi-definiteness and T-positive definiteness of
the T-quadratic form $F_{\mathcal{A}}$, defined in (1.1). This would not be
easy by using the eigentuples of ${\mathcal{A}}$, as even for real square
matrices, their eigenvalues may not be real. Thus, as in the matrix case, we
symmetrize the T-square tensor ${\mathcal{A}}$.

Let ${\mathcal{A}}\in\Re^{n\times n\times p}$ be a T-square tensor. We say
that ${\mathcal{A}}$ is T-symmetric if ${\mathcal{A}}={\mathcal{A}}^{\top}$.
T-symmetric tensors have been studied in [12]. We have the following
propositions.

###### Proposition 3.1

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$. Then
${\mathcal{A}}+{\mathcal{A}}^{\top}$ is a T-symmetric tensor. Then
${\mathcal{A}}$ is positive semidefinite (definite) if and only if the
T-symmetric tensor ${\mathcal{A}}+{\mathcal{A}}^{\top}$ is positive
semidefinite (definite).

Proof Since

$\left({\mathcal{A}}+{\mathcal{A}}^{\top}\right)^{\top}={\mathcal{A}}^{\top}+\left({\mathcal{A}}^{\top}\right)^{\top}={\mathcal{A}}+{\mathcal{A}}^{\top},$

${\mathcal{A}}+{\mathcal{A}}^{\top}$ is T-symmetric.

For $X\in\Re^{n\times p}$, regard it as a tensor ${\mathcal{X}}\in\Re^{n\times
1\times p}$. We have

$F_{\mathcal{A}}(X)={\mathcal{X}}^{\top}*{\mathcal{A}}*{\mathcal{X}}=({\mathcal{X}}^{\top}*{\mathcal{A}}*{\mathcal{X}})^{\top}={\mathcal{X}}^{\top}*{\mathcal{A}}^{\top}*{\mathcal{X}}={1\over
2}{\mathcal{X}}^{\top}*({\mathcal{A}}+{\mathcal{A}}^{\top})*{\mathcal{X}}.$

Thus, ${\mathcal{A}}$ is positive semidefinite (definite) if and only if the
T-symmetric tensor ${\mathcal{A}}+{\mathcal{A}}^{\top}$ is positive
semidefinite (definite). . $\Box$

We thus study the eigentuples of T-symmetric tensors, and use them to analyze
positive semidefiniteness (definiteness) of these tensors.

The following proposition holds obviously.

###### Proposition 3.2

A T-square tensor ${\mathcal{A}}\in\Re^{n\times n\times p}$ is T-symmetric if
and only if bcirc$({\mathcal{A}})$ is symmetric. A T-square tensor
${\mathcal{A}}\in\Re^{n\times n\times p}$ is invertible if and only if
bcirc$(A)$ is invertible. In this case, we have

${\mathcal{A}}^{-1}={\rm bcirc}^{-1}({\rm bcirc}({\mathcal{A}}^{-1})).$

Furthermore, ${\mathcal{A}}$ is orthogonal in the sense of [1, 3, 4] if and
only if bcirc$({\mathcal{A}})$ is orthogonal.

We have the following theorem.

###### Theorem 3.3

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-symmetric
tensor. Then there are orthogonal tensor $\mathcal{U}\in\Re^{n\times n\times
p}$ and T-symmetric f-diagonal tensor $\mathcal{D}\in\Re^{n\times n\times p}$
such that

${\mathcal{A}}=\mathcal{U}*\mathcal{D}*\mathcal{U}^{\top}.$ (3.3)

Let the frontal slices of $\mathcal{D}$ be $D^{(1)},\cdots,D^{(p)}$. If
$\hat{\mathcal{D}}\in\Re^{n\times n\times p}$ is another T-symmetric
f-diagonal tensor, whose frontal slices $\hat{D}^{(1)},\cdots,\hat{D}^{(p)}$
are resulted from switching some diagonal elements of
$D^{(1)},\cdots,D^{(p)}$, then there is an orthogonal tensor
$\hat{\mathcal{U}}\in\Re^{n\times n\times p}$, such that

${\mathcal{A}}=\hat{\mathcal{U}}*\hat{\mathcal{D}}*\hat{\mathcal{U}}^{\top}.$
(3.4)

Proof Block circulant matrices can be block diagonalized with normalized
discrete Fourier transformation (DFT) matrix, which is unitary. Then, as in
(3.1) of [4], we have

$(F_{p}\otimes I_{n})\cdot{\rm bcirc}({\mathcal{A}})\cdot(F_{p}^{*}\otimes
I_{n})={\rm diag}(D_{1},\cdots,D_{p}),$ (3.5)

where $F_{p}$ is the $p\times p$ DFT matrix, $F_{p}^{*}$ is its conjugate
transpose, $\cdot$ is the standard matrix multiplication, $\otimes$ denotes
the Kronecker product. Since bcirc$({\mathcal{A}})$ is symmetric, by taking
conjugate transpose of (3.5), we see that $D_{1},\cdots,D_{p}$ in (3.1) of [4]
are all hermite. Applying the eigen-decomposition of
$D_{i}=U_{i}\Sigma_{i}U_{i}^{\top}$ for $i=1,\cdots,p$, we have

${\rm diag}(D_{1},\cdots,D_{p})={\rm diag}(U_{1},\cdots,U_{p}){\rm
diag}(\Sigma_{1},\cdots,\Sigma_{p}){\rm
diag}(U_{1}^{\top},\cdots,U_{p}^{\top}).$ (3.6)

Apply $(F_{p}^{*}\otimes I_{n})$ to the left and $(F_{p}\otimes I_{n})$ to the
right of each of the block diagonal matrices in (3.6). In each of the three
cases, the resulting triple product results in a block circulant matrix. We
have

${\rm bcirc}({\mathcal{A}})={\rm bcirc}(\mathcal{U}){\rm
bcirc}(\mathcal{D}){\rm bcirc}(\mathcal{U}^{\top}).$

This implies (3.3). Then we have

$\mathcal{D}=\mathcal{U}^{\top}*{\mathcal{A}}*\mathcal{U},$

and

$\mathcal{D}^{\top}=\left(\mathcal{U}^{\top}*{\mathcal{A}}*\mathcal{U}\right)^{\top}=\mathcal{U}^{\top}*{\mathcal{A}}^{\top}*\mathcal{U}=\mathcal{U}^{\top}*{\mathcal{A}}*\mathcal{U}=\mathcal{D},$

as ${\mathcal{A}}$ is T-symmetric. Thus, $\mathcal{D}$ is also T-symmetric.

Switching the order of eigenvalues in the eigen-decomposition
$D_{i}=U_{i}\Sigma_{i}U_{i}^{\top}$ for $i=1,\cdots,p$, we have (3.4). .
$\Box$

We call (3.3) a T-eigen-decomposition (TED) of ${\mathcal{A}}$.

###### Corollary 3.4

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-symmetric tensor
and (3.3) holds. Denote ${\mathcal{A}}^{*2}={\mathcal{A}}*{\mathcal{A}}$ and
${\mathcal{A}}^{*k}={\mathcal{A}}^{*(k-1)}*{\mathcal{A}}$ for any integer
$k\geq 3$. Then for any positive integer $k$, ${\mathcal{A}}^{*k}$ is still
T-symmetric, and we have

${\mathcal{A}}^{*k}=\mathcal{U}*\mathcal{D}^{*k}*\mathcal{U}^{\top}.$

###### Corollary 3.5

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-symmetric tensor
and (3.3) holds. Then ${\mathcal{A}}^{-1}$ exists if and only if
$\mathcal{D}^{-1}$ exists. If they exist, then they are T-symmetric and

${\mathcal{A}}^{-1}=\mathcal{U}*\mathcal{D}^{-1}*\mathcal{U}^{\top}.$

We may rewrite (3.3) as

${\mathcal{A}}*\mathcal{U}=\mathcal{U}*\mathcal{D},$ (3.7)

or

${\rm bcirc}({\mathcal{A}}){\rm bcirc}(\mathcal{U})={\rm
bcirc}(\mathcal{U}){\rm bcirc}(\mathcal{D}),$ (3.8)

Denote the $j$th lateral slice of $\mathcal{U}$ by $U_{j}\in\Re^{n\times p}$
for $j=1,\cdots,n$. Consider the $j$th column of (3.8) for $j=1,\cdots,n$. Let
$\mathcal{D}=(d_{ijk})$. Then $d_{ijk}=0$ if $i\not=j$. Let $d_{11k}\geq
d_{22k}\geq\cdots\geq d_{nnk}$ for $k=1,\cdots,p$. We have

${\mathcal{A}}*U_{j}={\bf d}_{j}\circ U_{j},$ (3.9)

where ${\bf d}_{j}=(d_{jj1},d_{jjp},d_{jj(p-1)},\cdots,d_{jj2})^{\top}$. Since
$\mathcal{U}$ is orthogonal, $U_{j}\not=O$. Thus, ${\bf d}_{j}$ is an
eigentuple of ${\mathcal{A}}$ with an eigenmatrix $U_{j}$.

For a matrix $U\in\Re^{n\times p}$, let its column vectors are ${\bf
u}^{(1)},\cdots,{\bf u}^{(p)}$. Then

$U=({\bf u}^{(1)},{\bf u}^{(2)},\cdots,{\bf u}^{(p)}).$

Denote $U^{[0]}=U$,

$U^{[1]}=({\bf u}^{(p)},{\bf u}^{(1)},\cdots,{\bf u}^{(p-1)}),$ $U^{[2]}=({\bf
u}^{(p-1)},{\bf u}^{(2)},\cdots,{\bf u}^{(p-1)}),$ $\cdots,$ $U^{[p-1]}=({\bf
u}^{(2)},{\bf u}^{(3)},\cdots,{\bf u}^{(1)}).$

Consider the $(n+j)$th column of (3.8). We have

${\mathcal{A}}*U_{j}^{[1]}={\bf d}_{j}\circ U_{j}^{[1]}.$ (3.10)

Thus, $U_{j}^{[1]}$ is also an eigenmatrix of ${\mathcal{A}}$, associated with
the eigentuple ${\bf d}_{j}$. Similarly, $U_{j}^{[2]},\cdots,U_{j}^{[p-1]}$
are also eigenmatrices of ${\mathcal{A}}$, associated with the eigentuple
${\bf d}_{j}$.

Consider the set of eigenmatrices

$T=\left\\{U_{j}^{[k]}:j=1,\cdots,n,k=0,\cdots,p-1\right\\}.$

Then $T$ forms an orthonormal basis of $\Re^{n\times p}$. For any two distinct
members $W,V\in T$, let $\mathcal{W}$ and $\mathcal{V}$ be the corresponding
$n\times 1\times p$ tensors. Then we have

$\mathcal{W}^{\top}*\mathcal{W}=\mathcal{I}_{11p},$ (3.11)

and

$\mathcal{W}^{\top}*\mathcal{V}={\mathcal{O}}_{11p}.$ (3.12)

Viewing (3.4), we may switch the order in
$\\{d_{11k},d_{22k},\cdots,d_{nnk}\\}$ for any $k=1,\cdots,p$. The resulted
$\hat{\bf d}_{j}$, $j=1,\cdots,n$ are still eigentuples of ${\mathcal{A}}$.
Hence, the number of eigentuples of ${\mathcal{A}}$ is large. But we may
always take ${\bf d}_{1},\cdots,{\bf d}_{n}$ in its standard form.

Combining the orthogonality of $\mathcal{U}$, we have the following theorem.

###### Theorem 3.6

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-symmetric
tensor. Then ${\mathcal{A}}$ has real eigentuples ${\bf d}_{1},\cdots,{\bf
d}_{n}$, such that ${\bf d}_{1}\geq{\bf d}_{2}\cdots\geq{\bf d}_{n}$. For each
$j$, $j=1,\cdots,n$, there are real eigenmatrices
$U_{j}^{[0]},\cdots,U_{j}^{[p-1]}$, of ${\mathcal{A}}$, associated with the
eigentuple ${\bf d}_{j}$. These $np$ eigenmatrices form an orthonormal basis
of $\Re^{n\times p}$.

We call the eigentuples $\\{{\bf d}_{1},\cdots,{\bf d}_{n}\\}$, satisfying
${\bf d}_{1}\geq{\bf d}_{2}\cdots\geq{\bf d}_{n}$, in Theorem 3.6 the set of
the principal eigentuples of ${\mathcal{A}}$.

If ${\mathcal{A}}=\mathcal{I}_{nnp}$, then
$\mathcal{U}=\mathcal{D}=\mathcal{I}_{nnp}$. Therefore, ${\bf
d}_{1}=\cdots={\bf d}_{n}=(1,0,\cdots,0)^{\top}$. If ${\mathcal{A}}$ has a set
of principal eigentuples ${\bf d}_{j}=(d_{j1},\cdots,d_{jp})^{\top}$ for
$j=1,\cdots,n$, then ${\mathcal{A}}+\lambda\mathcal{I}_{nnp}$ has a set of
principal eigentuples ${\bf
d}_{j}=(d_{j1}+\lambda,\cdots,d_{jp}+\lambda)^{\top}$ for $j=1,\cdots,n$.

We are not sure whether all eigentuples of a T-symmetric tensor are real and
two eigenmatrices associated with two distinct eigentuples of a T-symmetric
tensor are orthogonal to each other. However, but we can prove the following
theorem.

###### Theorem 3.7

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-symmetric
tensor, and $\\{{\bf d}_{1},\cdots,{\bf d}_{n}\\}$ is a set of principal
eigentuples of ${\mathcal{A}}$ such that ${\bf d}_{1}\geq\cdots\geq{\bf
d}_{n}$. Then for any real eigentuple ${\bf d}_{0}$ of ${\mathcal{A}}$, we
have

${\bf d}_{1}\geq{\bf d}_{0}\geq{\bf d}_{n}.$ (3.13)

Proof Assume that there is an eigenmatrix $V\in\mathbb{C}^{n\times p}$ such
that

${\mathcal{A}}*V={\bf d}_{0}\circ V.$

Taking conjugate, we have

${\mathcal{A}}*\bar{V}={\bf d}_{0}\circ\bar{V}.$

Let $W=V+\bar{V}$. Then $W$ is real and

${\mathcal{A}}*W={\bf d}_{0}\circ W.$

If $W$ is nonzero, then $U$ is a real eigenmatrix of ${\mathcal{A}}$,
associated with ${\bf d}_{0}$. Otherwise, $V$ is pure imaginary. Letting
$\hat{W}=\sqrt{-1}V$, we still have a real eigenmatrix of ${\mathcal{A}}$,
associated with ${\bf d}_{0}$. Without loss of generality, assume $W$ is such
a real eigenmatrix.

Let $U_{j}^{[0]},\cdots,U_{j}^{[p-1]}$ be the eigenmatrices of ${\mathcal{A}}$
in Theorem 3.6. Then we have real coefficients
$\alpha_{j}^{[0]},\cdots,\alpha_{j}^{[p-1]}$, for $j=1,\cdots,n$, such that

$W=\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}U_{j}^{[k-1]}.$

Let $\mathcal{U}_{j}^{[k]}$ be the $n\times 1\times p$ tensors corresponding
to $U_{j}^{[k]}$ for $j=1,\cdots,n$ and $k=0,\cdots,p-1$. Let $\mathcal{W}$ be
the $n\times 1\times p$ tensor corresponding to $W$. Let $\mathcal{D}_{j}$ be
the $1\times 1\times p$ tensors corresponding to ${\bf d}_{j}$ for
$j=0,\cdots,n$. Then

$\displaystyle\mathcal{W}^{\top}*{\mathcal{A}}*\mathcal{W}$ $\displaystyle=$
$\displaystyle\mathcal{W}^{\top}*\mathcal{W}*\mathcal{D}_{0}$ $\displaystyle=$
$\displaystyle\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\mathcal{U}_{j}^{[k-1]}\right)^{\top}*\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\mathcal{U}_{j}^{[k-1]}\right)*\mathcal{D}_{0}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}\mathcal{I}_{11p}*\mathcal{D}_{0}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}\mathcal{D}_{0}.$

On the other hand,

$\displaystyle\mathcal{W}^{\top}*{\mathcal{A}}*\mathcal{W}$ $\displaystyle=$
$\displaystyle\mathcal{W}^{\top}*{\mathcal{A}}*\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\mathcal{U}_{j}^{[k-1]}\right)$
$\displaystyle=$
$\displaystyle\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\mathcal{U}_{j}^{[k-1]}\right)^{\top}*\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}{\mathcal{A}}*\mathcal{U}_{j}^{[k-1]}\right)$
$\displaystyle=$
$\displaystyle\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\mathcal{U}_{j}^{[k-1]}\right)^{\top}*\left(\sum_{j=1}^{n}\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\mathcal{U}_{j}^{[k-1]}*\mathcal{D}_{j}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}\mathcal{I}_{11p}*\mathcal{D}_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}\mathcal{D}_{j}.$

From this, we have

$\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}\mathcal{D}_{0}=\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}\mathcal{D}_{j},$

i.e.,

$\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}{\bf
d}_{0}=\sum_{j=1}^{n}\left(\sum_{k=1}^{p}\alpha_{j}^{[k-1]}\right)^{2}{\bf
d}_{j}.$

The inequality (3.13) is obtained. . $\Box$

###### Corollary 3.8

The eigentuples ${\bf d}_{1}$ and ${\bf d}_{n}$ are unique to ${\mathcal{A}}$.

We call ${\bf d}_{1}$ the largest eigentuple of ${\mathcal{A}}$, and ${\bf
d}_{n}$ the smallest eigentuple of ${\mathcal{A}}$.

##  4 T-Symmetric Positive Semidefinite Tensors

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-square tensor.
Then by Proposition LABEL:p3.1, the T-quadratic form is positive semidefinite
(definite) if and only if the T-symmetric tensor
${\mathcal{A}}+{\mathcal{A}}^{\top}$ is positive semidefinite (definite). This
stimulates us to study positive semidefiniteness (definiteness) of a
T-symmetric tensor.

###### Theorem 4.1

Suppose that ${\mathcal{A}}\in\Re^{n\times n\times p}$ is a T-symmetric tensor
and it has a set of principal eigentuples ${\bf d}_{1},\cdots,{\bf d}_{n}$,
such that ${\bf d}_{1}\geq{\bf d}_{2}\cdots\geq{\bf d}_{n}$. Then
${\mathcal{A}}$ is positive semidefinite (definite) if and only if the
smallest eigentuple ${\bf d}_{n}\geq(>){\bf 0}$.

Proof By Theorem 3.6, ${\bf d}_{1}\geq{\bf d}_{2}\cdots\geq{\bf d}_{n}$, and
for each $j$, $j=1,\cdots,n$, there are real eigenmatrices
$U_{j}^{[0]},\cdots,U_{j}^{[p-1]}$, of ${\mathcal{A}}$, associated with the
eigentuple ${\bf d}_{j}$, such that these $np$ eigenmatrices form an
orthonormal basis of $\Re^{n\times p}$.

If ${\bf d}_{n}$ is not nonnegative, let $U=U_{n}^{[0]}$ and
$\mathcal{U}_{n}^{[0]}$ be the corresponding $n\times 1\times p$ tensor. Let
$\L$ be the $1\times 1\times p$ tensor corresponding to ${\bf d}_{n}$. Then

$F_{\mathcal{A}}(U)=\left(\mathcal{U}_{n}^{[0]}\right)^{\top}*{\mathcal{A}}*\mathcal{U}_{n}^{[0]}=\left(\mathcal{U}_{n}^{[0]}\right)^{\top}*\mathcal{U}_{n}^{[0]}*\L=\mathcal{I}_{nnp}*\L\not\geq{\bf
0},$

which implies that $F_{\mathcal{A}}$ is not positive semi-definite. Similarly,
if ${\bf d}_{n}$ is not positive, then $F_{\mathcal{A}}$ is not positive
definite.

On the other hand, suppose that ${\bf d}_{n}\geq 0$. Let
$\mathcal{U}_{j}^{[k]}$ be the $n\times 1\times p$ tensor corresponding to
$U_{j}^{[k]}$ for $j=1,\cdots,n$ and $k=0,\cdots,p-1$. Let $X\in\Re^{n\times
p}$. Then there are real coefficients $\alpha_{j}^{[k]}$ for $j=1,\cdots,n$
and $k=0,\cdots,p-1$, such that

$X=\sum_{j=1}^{n}\sum_{k=0}^{p-1}\alpha_{j}^{[k]}U_{j}^{[k]}.$

Let $\L_{j}$ be the $1\times 1\times p$ tensor corresponding to ${\bf s}_{j}$
for $j=1,\cdots,n$. We have

$\displaystyle F_{\mathcal{A}}(X)$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{n}\sum_{l=0}^{p-1}\alpha_{i}^{[l]}\mathcal{U}_{i}^{[l]}\right)^{\top}*{\mathcal{A}}*\left(\sum_{j=1}^{n}\sum_{k=0}^{p-1}\alpha_{j}^{[k]}\mathcal{U}_{j}^{[k]}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{l=0}^{p-1}\sum_{k=0}^{p-1}\alpha_{i}^{[l]}\alpha_{j}^{[k]}(\mathcal{U}_{i}^{[l]})^{\top}*{\mathcal{A}}*\mathcal{U}_{j}^{[k]}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{l=0}^{p-1}\sum_{k=0}^{p-1}\alpha_{i}^{[l]}\alpha_{j}^{[k]}(\mathcal{U}_{i}^{[l]})^{\top}*\mathcal{U}_{j}^{[k]}*\L_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\sum_{k=0}^{p-1}\left(\alpha_{j}^{[k]}\right)^{2}{\bf
d}_{j}$ $\displaystyle\geq$ $\displaystyle{\bf 0}.$

Thus, $F_{\mathcal{A}}$ is positive semidefinite. Similarly, if ${\bf
d}_{n}\geq{\bf 0}$, then $\mathcal{F}_{\mathcal{A}}$ is positive definite. .
$\Box$

##  5 Relation with TSVD of General Third Order Tensors

Suppose that ${\mathcal{A}}\in\Re^{m\times n\times p}$. By [4],
${\mathcal{A}}$ has a T-singular value decomposition (TSVD):

${\mathcal{A}}=\mathcal{U}*\mathcal{S}*\mathcal{V}^{\top},$ (5.14)

where $\mathcal{U}\in\Re^{m\times m\times p}$ and $\mathcal{V}\in\Re^{n\times
n\times p}$ are orthogonal tensors, $\mathcal{S}\in\Re^{m\times n\times p}$ is
a f-diagonal tensor.

###### Theorem 5.1

Suppose that ${\mathcal{A}}\in\Re^{m\times n\times p}$ with TSVD (5.14). Then
${\mathcal{A}}*{\mathcal{A}}^{\top}\in\Re^{m\times m\times p}$ and
${\mathcal{A}}^{\top}*{\mathcal{A}}\in\Re^{n\times n\times p}$ are T-symmetric
positive semi-definite tensors with TED

${\mathcal{A}}*{\mathcal{A}}^{\top}=\mathcal{U}*(\mathcal{S}*\mathcal{S}^{\top})*\mathcal{U}^{\top},$

and

${\mathcal{A}}^{\top}*{\mathcal{A}}=\mathcal{V}*(\mathcal{S}^{\top}*\mathcal{S})*\mathcal{V}^{\top},$

respectively.

We now define singular tuples and singular matrices of general third order
tensors. Suppose that ${\mathcal{A}}\in\Re^{m\times n\times p}$ is a third
order tensor, $X\in\Re^{n\times p}$, $X\not=O$, $Y\in\Re^{m\times p}$,
$Y\not=O$, ${\bf s}\in\Re^{p}$, and

${\mathcal{A}}*X={\bf s}\circ Y$ (5.15)

and

${\mathcal{A}}^{\top}*Y={\bf s}\circ X.$ (5.16)

Then we call ${\bf s}$ a singular tuple of ${\mathcal{A}}$, $X$ a right
singular matrix of ${\mathcal{A}}$, $Y$ a right singular matrix of
${\mathcal{A}}$, corresponding to the singular tuple ${\bf s}$.

###### Theorem 5.2

Suppose that ${\mathcal{A}}\in\Re^{m\times n\times p}$ is a third order
tensor. Without loss of generality, assume that $n\leq m$. Then
${\mathcal{A}}$ has singular tuples ${\bf s}_{1}\geq{\bf
s}_{2}\geq\cdots\geq{\bf s}_{n}\geq{\bf 0}$. For each $j$, $j=1,\cdots,n$,
there are right singular matrices $U_{j}^{[0]},\cdots,U_{j}^{[p-1]}$, and left
singular matrices $V_{j}^{[0]},\cdots,V_{j}^{[p-1]}$, of ${\mathcal{A}}$,
associated with the singular tuple ${\bf s}_{j}$. The $np$ singular matrices
$U_{j}^{[0]},\cdots,U_{j}^{[p-1]}$ form an orthonormal basis of $\Re^{n\times
p}$, and the $np$ singular matrices $V_{j}^{[0]},\cdots,V_{j}^{[p-1]}$ form a
part of an orthonormal basis of $\Re^{m\times p}$, respectively.

Furthermore, ${\mathcal{A}}^{\top}*{\mathcal{A}}\in\Re^{n\times n\times p}$
and ${\mathcal{A}}*{\mathcal{A}}^{\top}\in\Re^{m\times m\times p}$ are two
T-symmetric positive semidefinite tensors. The tensor
${\mathcal{A}}^{\top}*{\mathcal{A}}$ has $n$ nonnegative eigentuples ${\bf
s}_{1}^{\odot 2},\cdots,{\bf s}_{n}^{\odot 2}$. For each $j$, $j=1,\cdots,n$,
there are real eigenmatrices $U_{j}^{[0]},\cdots,U_{j}^{[p-1]}$, of
${\mathcal{A}}$, associated with the eigentuple ${\bf s}_{j}^{\odot 2}$. The
tensor ${\mathcal{A}}*{\mathcal{A}}^{\top}$ has $n$ nonnegative eigentuples
${\bf s}_{1}^{\odot 2},\cdots,{\bf s}_{n}^{\odot 2}$. For each $j$,
$j=1,\cdots,n$, there are real eigenmatrices
$V_{j}^{[0]},\cdots,V_{j}^{[p-1]}$, of ${\mathcal{A}}$, associated with the
eigentuples ${\bf s}_{j}^{\odot 2}$. If $n<m$, for each $j$, $j=n+1,\cdots,m$,
there are real eigenmatrices $V_{j}^{[0]},\cdots,V_{j}^{[p-1]}$, of
${\mathcal{A}}$, associated with the zero eigentuple ${\bf
0}\in\mathbb{R}^{p}$. The $mp$ singular matrices
$V_{j}^{[0]},\cdots,V_{j}^{[p-1]}$ form an orthonormal basis of $\Re^{m\times
p}$.

Acknowledgment We are thankful to Prof. Yicong Zhou and Dr. Dongdong Chen for
the discussion on the multi-view clustering problem and the image feature
extraction problem.

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