Posts Tagged ‘Conferences’

LaTeX Poster Template

August 14th, 2009



LaTeX Poster


  • Four column landscape layout instead of three column portrait layout.
  • Messed around with the header.





Sylvester’s Law of Inertia and *-Congruence

August 7th, 2009

Recall for matrices that a similarity transformation is one that takes a (square) matrix A to SAS-1, where S is some invertible matrix of the same size as A. Standard linear algebra courses like to beat into students just how important similarity transformations are, which makes sense from their point of view because linear algebra is really more about studying operators than matrices themselves, and similarity transformations correspond to just a change of basis. The downside is that many people end up thinking of similarity as the equivalence relation among matrices and hence the Jordan canonical form as the canonical form. This month’s Lemma of the Month is about the canonical form under a different equivalence relation: *-congruence.

What is *-Congruence?

Two square matrices A and B are said to be *-congruent if there is an invertible matrix S such that SAS* = B. Even though no inverses are present in that equation, invertibility really is required for any nice results to be obtained (for example, *-congruence would not even define an equivalence relation if S were allowed to be singular). For people who like knowing applications of mathematical tools before knowing the tools themselves, *-congruence has an important role when dealing with quadratic forms. Also, the inertia of a matrix, which will be defined in the next paragraph, comes into play when looking at sign pattern matrices (apologies for the lack of a wiki link; it seems as though sign pattern matrices are at that ugly stage after there are lots of papers written about them but before there is a Wikipedia page written about them).

Sylvester’s Law of Inertia

By far the most well-known result about *-congruence is Sylvester’s Law of Inertia, which gets its name from the seldom-used inertia of a matrix. The inertia of a Hermitian matrix A is defined to be the tuple

Inertia of a matrix

where n+ is the number of positive eigenvalues of A, n0 is the number of zero eigenvalues of A, and n is the number of negative eigenvalues of A (recall that Hermitian matrices have real eigenvalues, so this definition makes sense). Sylvester’s Law of Inertia is as follows:

Theorem [1,2]. Let A, B ∈ Mn be Hermitian. Then A and B are *-congruent if and only if they have the same inertia.

One obvious corollary of this theorem is that every Hermitian matrix A is *-congruent to a diagonal matrix with n+ ones, n0 zeroes, and n negative ones on the diagonal; this is the canonical form for Hermitian matrices under *-congruence.

Generalizations of Sylvester’s Law

If you’re a linear algebra nerd like me, then you might be thinking to yourself that Sylvester’s Law of Inertia seems trivial in light of the Spectral Decomposition Theorem; for any Hermitian A we could find a unitary so that UAU* is real and diagonal, and then we could find a diagonal D such that DUAU*D* is scaled appropriately to satisfy the theorem. Setting S = DU then shows us that we can always arrive at the canonical form.

Notice, however, that the Spectral Decomposition Theorem applies not only to Hermitian matrices, but to normal matrices as well. Thus, using the same logic outlined in the previous paragraph, it stands to reason that the canonical form under *-congruence for normal matrices should be a diagonal matrix in which each diagonal entry either has modulus one or zero. Indeed, this is essentially the content of the following theorem of Ikramov, which was proved in 2001.

Theorem [3]. Let A, B ∈ Mn be normal. Then A and B are *-congruent if and only if they have the same number of eigenvalues on each open ray from the origin in the complex plane.

This theorem truly generalizes Sylvester’s Law of Inertia; in the Hermitian case, the only two open rays that eigenvalues can lie on are the positive real line and the negative real line. What I would like to know, though, is why on Earth this theorem was first proved/published so recently — it seems like such a natural generalization of Sylvester’s Law of Intertia that it seems like it should be a very old result! Is it just a hole in the literature that wasn’t noticed until recently?

Anyway, for the really interested reader, there is a generalization of these results to the case when A and B need not even be normal, but it’s quite technical so I won’t present it here. See [4] if you’re curious.

Thanks to Roger Horn for his great series of lectures at the Summer School and Advanced Workshop on Trends and Developments in Linear Algebra, which introduced me to the generalizations of Sylvester’s Law.


  1. Sylvester, J. J., A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. Philosophical Magazine IV: 138–142 (1852).
  2. Horn, R. and Johnson, C., Matrix analysis. Cambridge University Press (1990).
  3. Ikramov, Kh. D., On the inertia law for normal matrices. Doklady Math. 64 (2001) 141-142.
  4. Horn, R. and Sergeichuk, V., Canonical forms for complex matrix congruence and *congruence. Linear Algebra Appl. 416 (2006) 1010-1032. arXiv:0709.2473v1 [math.RT]

A Backward Triangle Inequality for Matrices

July 3rd, 2009

This month’s Lemma of the Month comes all the way from the Summer School and Advanced Workshop on Trends and Developments in Linear Algebra in Trieste, Italy, where Professor Rajendra Bhatia presented a lecture that introduced several simple yet endlessly interesting matrix inequalities. I will briefly present the various results here without proof, though the proof of the first “stepping stone” lemma is provided in the PDF attached to the bottom of this post. The truly interested reader can find full proofs in Professor Bhatia’s notes (follow the link above) or in [1].

Recall that one of the defining properties of a matrix norm is that it satisfies the triangle inequality:

Triangle Inequality

So what can we say about generalizing the backward triangle inequality to matrices? We can of course replace A by A – B in the above equation to find the following backward triangle inequality:

Backward Triangle Inequality

However, what happens if we swap the roles of the absolute value and the matrix norm on the left-hand side? That is, if we recall that |A| is the positive semidefinite part of A (i.e., the square root of A*A), then can we say something like

Incorrect Backward Triangle Inequality

It turns out that the answer to this question is heavily norm-dependent, so we will focus on the norm that gives the simplest answer: the Frobenius norm, which I will denote by ||•||2.

Theorem [Araki-Yamagami]. Let A, B ∈ Mn. Then


Finally, it just wouldn’t seem right to post from Italy without sharing a bit of it. Thus, I leave you with a taste of the highlight of the trip (excepting the mathematics, of course).

Building Up to the Result

In order to prove the result, one can proceed by proving a series of simple commutant-type matrix norm inequalities, which are interesting in their own right.

Lemma. Let A, B ∈ Mn be positive semi-definite and let X ∈ Mn be arbitrary. Then

Lemma 1

Lemma. Let A ∈ Mn be Hermitian. Then

Lemma 2

Lemma. Let A, B ∈ Mn be Hermitian. Then

Lemma 3


The colosseum as of seven hours ago

The colosseum as of eight hours ago

Related Links


  1. H. Araki and S. Yamagami, An inequality for the Hilbert-Schmidt norm, Commun. Math. Phys., 81 (1981) 89-98.

TQC 2009

May 13th, 2009

I spent the beginning of this week at the 4th Workshop on Theory of Quantum Computation, Communication and Cryptography, and I decided that I might as well post some information and thoughts about it since it was probably the best quantum workshop that I’ve attended thus far (not least because it was only three days long and the Institute for Quantum Computing knows how to do food right). There were some fifteen or so talks, and here I’ll describe two of my personal favourites.

Quantum algorithm for online memory checking

Wim van Dam and Qingqing Yuan

In his talk on online memory checking, Wim van Dam described the problem of trying to retrieve data from an unreliable and public server. Because the server is public, an adversary could tamper with its data, rendering the information that you retrieve useless. The online memory checking question of interest is then how to make sure that the requested information has not been tampered with.

One obvious (albeit largely useless) approach to making sure that the public server has not been tampered with is to simply keep a copy of its entire contents on a private server as well. When data is requested from the public server, check to see if it matched the data on the private server, and you’re good to go. This carries the obvious problem that you have to keep two copies of all of the data.

The obvious improvement to this is to use some sort of hashing function on the contents of the public server, store the results on a private server, and then check to see if the hash of your retrieved data equals the hash of the stored data whenever you access the public server. The question of how large the hash (or whatever encoding of the information you use) must be was answered in 1994 by Blum et al. and in 2005 by Naor and Rothblum: s × t ∈ Ω(n), where n is the size of the data on the public server, s is the size of the data stored on the private server, and t is the number of queries made to the public server.

Wim van Dam and Qingqing Yuan show that if the memory checker is allowed to act in a quantum mechanical manner, then the complexity of the problem is reduced to s × t ∈ Ω(polylog(n)). Score 1 for quantum algorithms.

A necessary condition for approximate quantum error correction

Cedric Bény

It’s not particularly surprising that this talk spoke to me pretty directly, as it deals with the same approach to the same subarea of the same field of research that I do. That and my name was mentioned in the talk. Score 10 for Cedric Bény.

The main result of the talk was a generalization of the Knill-Laflamme conditions to the realm of approximate quantum error correction (as opposed to perfect quantum error correction).  The idea was that the Knill-Laflamme conditions can be restated in terms of the complementary channel in a very natural way: a given subspace is correctable for a channel if and only if the complementary channel sends everything supported on that subspace to a single fixed operator (another way of saying this is that the subspace is private for the complementary channel, a connection that was explored by Kretschmann, Kribs and Spekkens in 2007).

It turns out that there is a natural way to generalize this formulation of the Knill-Laflamme conditions. The idea is that if the complementary channel, when restricted to a given subspace, is within a small distance of a channel that maps everything to a fixed operator (where distance is measured via the diamond norm), then that subspace is approximately correctable (and he of course quantifies the “small distance” and “approximately”).

He also provides a generalization of these results to approximate error correction of subsystem codes, which is a nice bonus.  Finally, it’s interesting to note that the set of operators that are approximately correctable by a given recovery operation does not necessarily form an algebra, in contrast to the perfect error correction situation.