Singular Value Decomposition at FIFA 2022
Peter Zizler and Mandana Sobhanzadeh
Mount Royal University, Calgary

###### Abstract

The singular value decomposition is arguably one of the most fundamental
results in linear algebra. While rigorous proof of this result is of
importance, equally important is the motivation in the applied settings. We
provide a lively and quite intuitive presentation on the appearance of the
singular value decomposition of a matrix in the round robin tournaments, as
well as the polar decomposition, all in the context of FIFA 2022 World Cup.
This exposition is intended to be implemented in a class setting or given as a
take home project in the second linear algebra course. Given the popularity of
this world event we expect students to be interested and engaged throughout
the analysis and ensuing conversations. Discussions among students should be
enriching as they attempt to analyse the various FIFA groups and give
comparisons. Furthermore, ideas in regards how to best interpret the singular
vectors should be quite interesting. Basic notions and results from the first
year linear algebra course are assumed.

##  1 Introduction

Our mathematical essay can be seen as a motivation to learn the need for the
singular value decomposition of a matrix as well as its closely related
result, the polar decomposition of a matrix. We will see how the singular
value decompsotion of a matrix has a practical meaning in round robin
tournaments. Moreover, students will also appreciate and discuss the closely
related polar decomposition of a matrix, a result that will find a specific
interpretation in the round robin tournament matrix as well.

Given the results of the round robin tournament it is natural to seek the
corresponding teams’ offense and defense scores. As a result we have a
motivation for the rank one matrix approximation, the tensor product
approximation, of the tournament matrix $A$. It is intuitive that the team’s
offense and defense ability can be estimated by goal counts, for and against,
in round robin tournaments. Students will be encouraged to explore other
options, possibly weighted goal counts, both for and agaisnt. The choice of
the appropriate wieights might stimulate enriching discusions that will
enhance future deeper understadning the the forthcoming singular value
decomposition of $A$, in particular the interpretation of its dominant
singular vectors. Students will find the offense and defense scores for the
teams, stemming out of the singular value decomposition of a matrix, are
sensitive to goal scoring on good or bad defensive teams, and goals against
from good or bad offense teams. In matrix theory language, we motivate the
best least squares approximation of $A$ by a rank one matrix (tensor),

$A\approx\sigma_{1}{\bf u}_{1}{\bf v}_{1}^{T}$

where ${\bf u}_{1}$ are the normalized team offense scores and ${\bf v}_{1}$
are the normalized team defense scores, $\sigma_{1}$ being a scaling factor.

We continue to analyze all groups at FIFA 2022 using these techniques with
further finer analysis of team interactions in the tournaments based on the
other rank one approximations in the singular value decomposition. Connecting
our exposition to a popular event like the FIFA 2022 World Cup can motivate
the students to engage and sustain interest. On the other hand, these methods
can be applied in any round robin tournament for any relevant sport activities
should the students seek their favourite sports settings.

During the FIFA World Cup in Qatar, Fall $2022$, a very interesting scenario
arose when Argentina, Poland, Mexico and Saudi Arabia played round robin
games, from which two teams would advance. This was the group C at the soccer
tournament. If a team wins against another team the winning team gets awarded
three points, the team that lost gets zero points. If the match results in a
tie then both teams get awarded one point. The two teams with the most points
advance. However, many teams might have the same number of points, in fact,
this happens relatively often. In such an event, the goal difference then
decides. One collects all the goals scored by the team (in the three matches)
minus the goals conceded by the team (in the three matches). If this still
fails to decide, then only the number of goals scored by the team in the three
matches decide without any consideration to the conceded goals by the team. If
this does not decide, then the team head to head results decide who advances.
If this still does not decide then the teams with fewer disciplinary actions
against them advance, a rule that involves yellow and red cards. This is
referred to as the fair play rule. Finally, if this still does not decide,
then a toss of a coin will.

Argentina had $6$ points, Poland had $4$ points, Mexico had $4$ points and
Saudi Arabia had $3$ points. Argentina would advance, Saudi Arabia was in the
last place. The decision had to be made between Poland and Mexico. Poland goal
differential was $0$ where as Mexico goal diffential was $-1$. Poland
advanced. However, the excitement at the time was the following. Argentia
played Poland and Mexico played Saudi Arabia as the last two games in the
group simultaneously. Argentina lead $2:0$ and Mexico lead $2:0$ to the very
last minutes of the games. If the score remained as such then the decision
would had to be made by the fair play rule. However, Saudi Arabia scored.

The results of the round robin tournament reflect the teams’ offensive ability
as well as their defensive ability. As the teams play in the tournament they
score goals against other teams during the matches. The total amount of goals
scored by a team in the tournament is a possible way to measure the team’s
offensive ability, however it values each goal scored equally, goal against a
good defensive team in the same way as a goal against a poor defensive team.
It is harder to score against defensive teams. Similarly, each team has a
certain amount of goals scored on them. The count of goals against is a
possible measure for the team’s defensive ability, however, it is easier to
defend against a poor offensive team as opposed to a good offensive team.
Offense and defense scores for the teams, which are sensitive to this aspect,
can be obtained from the singular value decomposition of a matrix. The matrix
in question is the offense, defense performance matrix associated with the
given round robin tournament.

We will assign normalized offense scores to each team in the tournament as
well as the normalized defense scores for each team. This will in turn induce
the offense and defense rankings for the teams in the tournament. Compared to
the offense and defense rankings stemming out of total goals scored in the
tournament, and goals scored against in the tournament, the singular value
decomposition rankings break ties that often arise in the tournament. Many
times the scores from the singular value decompostion rankings, offense and
defense, yield the same ranking results as the total goals count, however
there are cases, some groups at FIFA 2022, where the rankings obtained are
different. We will show these results in our paper. We will analyze each group
at the 2022 FIFA world cup, groups A through H. We first illustrate the
technique on the group C. The techniques are applicable to any round robin
tournament. The analysis drawn can be utilized to predict the team scoring
outcomes should the teams play again under the same attributes. The ideas
appearing in the paper are inspired by the work of [3]. For more works on
related topics see [2]. We now present the singular value decomposition, the
offense, defense decomposition for the group C at FIFA 2022.

##  2 The Singular Value Decomposition

The round robin results for the group C at FIFA 2022 were

| Argentina | Poland | Mexico | Saudi Arabia
---|---|---|---|---
Argentina | x | 2:0 | 2:0 | 1:2
Poland | 0:2 | x | 0:0 | 2:0
Mexico | 0:2 | 0:0 | x | 2:1
Saudi Arabia | 2:1 | 0:2 | 1:2 | x

We consider the results in this tournament using the lenses of the team’s
offense and defense ability. What counts now are the goals scored by the teams
and against, winning, tying or losing a game is irrelevant. To this end, we
form the offense defense performance matrix $A$ reflecting the tournament
results

$A=\left(\begin{array}[]{cccc}1.1667&2.00&2.00&1.00\\\ 0&0.6667&0&2.00\\\
0&0&0.8333&2.00\\\ 2.00&0&1.00&1.3333\\\ \end{array}\right)$

The off diagonal value $A(i,j)$, $i\neq j$, is the number of goals scored by
the team $i$ against the team $j$. However, lively discussions in regards to
the diagonal entires in the matrix $A$ should ensue among students. It
reflects a hypothetical scenario where a team plays against itself and it
should capture both the offense and the defensive ability of the team. One
option is to set the diagonal value $A(i,i)$ to be the mean of all row $i$ and
column $i$ entries, excluding the diagonal, all $6$ of them. The diagonal
entry $A(i,i)$ thus averages out the goals scored by the team $i$ and goals
scored against the team $i$. Having said that, we presume that students might
suggest further ideas on this. We consider the singular value decomposition of
$A$

$\displaystyle A$ $\displaystyle=$ $\displaystyle UDV^{T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0.6652&0.5771&-0.4710&0.0512\\\
0.3629&-0.5899&-0.2826&-0.6636\\\ 0.4038&-0.5547&-0.0304&0.7269\\\
0.5127&0.1057&0.8351&-0.1692\\\ \end{array}\right)$
$\displaystyle\left(\begin{array}[]{cccc}4.3328&0&0&0\\\ 0&2.1135&0&0\\\
0&0&1.5973&0\\\
0&0&0&0.6971\end{array}\right)\left(\begin{array}[]{cccc}0.4158&0.4186&0.7016&-0.3997\\\
0.3629&0.3600&-0.7078&-0.4877\\\ 0.5030&0.3774&-0.0828&0.7731\\\
0.6652&-0.7435&0.0102&-0.0687\\\ \end{array}\right)$

$\displaystyle A$ $\displaystyle=$ $\displaystyle\sigma_{1}{\bf u}_{1}{\bf
v}^{T}_{1}+\sigma_{2}{\bf u}_{2}{\bf v}^{T}_{2}+\sigma_{3}{\bf u}_{3}{\bf
v}^{T}_{3}+\sigma_{4}{\bf u}_{4}{\bf v}^{T}_{4}$ $\displaystyle=$
$\displaystyle A_{1}+A_{2}+A_{3}+A_{4}$ $\displaystyle=$ $\displaystyle
4.3328\left(\begin{array}[]{c}0.6652\\\ 0.3629\\\ 0.4038\\\
0.5127\end{array}\right)\left(\begin{array}[]{cccc}0.4158&0.3629&0.5030&0.6652\\\
\end{array}\right)$ $\displaystyle+$ $\displaystyle
2.1135\left(\begin{array}[]{c}0.5771\\\ -0.5899\\\ -0.5547\\\ 0.1057\\\
\end{array}\right)\left(\begin{array}[]{cccc}0.4186&0.3600&0.3774&-0.7435\\\
\end{array}\right)$ $\displaystyle+$ $\displaystyle
1.5973\left(\begin{array}[]{c}-0.471010\\\ -0.282612\\\ -0.030440\\\
0.835077\\\
\end{array}\right)\left(\begin{array}[]{cccc}0.701562&-0.707705&-0.082831&0.010201\\\
\end{array}\right)$ $\displaystyle+$ $\displaystyle
0.6971\left(\begin{array}[]{c}0.051229\\\ -0.663640\\\ 0.726862\\\
-0.169204\\\
\end{array}\right)\left(\begin{array}[]{cccc}-0.399693&-0.487699&0.773091&-0.068748\\\
\end{array}\right)$

In particular,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1.1982&1.0458&1.4497&1.9170\\\
0.6537&0.5705&0.7909&1.0458\\\ 0.7274&0.6348&0.8800&1.1637\\\
0.9236&0.8061&1.1174&1.4776\\\
\end{array}\right)+\left(\begin{array}[]{cccc}0.5106&0.4392&0.4604&-0.9069\\\
-0.5219&-0.4489&-0.4706&0.9270\\\ -0.4907&-0.4221&-0.4425&0.8716\\\
0.0935&0.0804&0.0843&-0.1660\\\ \end{array}\right)$ $\displaystyle+$
$\displaystyle\left(\begin{array}[]{cccc}-0.5278&0.5325&0.0623&-0.0077\\\
-0.3167&0.3195&0.0374&-0.0046\\\ -0.0341&0.0344&0.0040&-0.0005\\\
0.9358&-0.9440&-0.1105&0.0136\\\
\end{array}\right)+\left(\begin{array}[]{cccc}-0.0143&-0.0174&0.0276&-0.0025\\\
0.1849&0.2256&-0.3577&0.0318\\\ -0.2025&-0.2471&0.3917&-0.0348\\\
0.0471&0.0575&-0.0912&0.0081\\\ \end{array}\right)$

The vectors $\left\\{{\bf v}_{i}\right\\}^{4}_{i=1}$ form an orthonormal basis
for ${\bf R}^{4}$, the same holds for the set $\left\\{{\bf
u}_{i}\right\\}^{4}_{i=1}$. We have $A{\bf v}_{i}=\sigma_{i}{\bf u}_{i}$ for
$i\in\\{1,2\ldots,4\\}$. The matrix $A_{1}$ is the best (in least squares
sense) rank one matrix approximating the matrix $A$. In particular,

${\rm min}\left\\{||A-B||_{2}\mbox{ where }B\mbox{ is rank one matrix
}\right\\}=||A-A_{1}||_{2}$

here $||C||^{2}_{2}$ denotes the square of the Frobenius norm of the matrix
$C$, the sum of the squares of the absolute values of the entries in the
matrix $C$. The matrix $A_{2}$ is the best (in least squares sense) rank one
matrix approximating the matrix $A-A_{1}$. We continue this recursively and
the procedure ends after four steps. For a very nice exposition of the
singular value decomposition we refer the reader to [5]. The singular vectors
${\bf u}_{1}$ and ${\bf v}_{1}$ corresponing to the largest singular value
$\sigma_{1}$ of $A$ are guaranteed to have nonnegative entries, due to the
Perron - Frobenius Theorem, since the entries in the matrix $A^{T}A$ as well
as $AA^{T}$ are all nonnegative. For a good reference on the Perron-Frobenius
Theorem see [4].

###  2.1 The Offense and Defense Scores

The (normalized) offense scores are the entries in the vector
${\bf u}_{1}=(0.6652,0.3629,0.4038,0.5127)^{T}$. In particular, Argentina is
awarded the highest score of $0.6652$, then Saudi Arabia with score of
$0.5127$, then Mexico with $0.4038$ and Poland last with $0.3629$. The higher
the offense score the better the offense rating. The (normalized) defense
scores are the entries in the vector ${\bf
v}_{1}=(0.4158,0.3629,0.5030,0.6652)^{T}$, higher the score worse the defense.
In the defense scores Poland is the best with $0.3629$, then Argentina with
$0.4158$, then Mexico with $0.5030$ and Saudi Arabia last with $0.6652$. We
note that normalized here means the sum of the squares of the entries in the
vector is unity. The best rank one approximation for $A$ in the least squares
sense is given by $A_{1}=\sigma_{1}{\bf u}_{1}{\bf v}^{T}_{1}$. The singular
value decomposition scores yield the following ranking. Offense: Argentina,
Saudi Arabia, Mexico, Poland. Defense: Poland, Argentina, Mexico, Saudi
Arabia. We note the total goal count yields the same ranking, albeit induces
ties. In particular Argentina scored $5$ goals, Poland $2$ goals, Mexico $2$
goals, and Saudi Arabia $3$ goals. We define the goal scored vector
$(5,2,2,3)^{T}$ and upon normalization obtain the vector
$(0.7715,0.3086,0.3086,0.4629)^{T}$. This vector can be seen as a crude
approximation of the singular vector ${\bf u}_{1}$. Argentina allowed $2$
goals, Poland $2$ goals, Mexico $3$ goals, and Saudi Arabia $5$ goals. We
define the goal allowed vector $(2,2,3,5)^{T}$ and upon normalization obtain
the vector $(0.3086,0.3086,0.4629,0.7715)^{T}$. This vector can be seen as a
crude approximation of the singular vector ${\bf v}_{1}$. For more work on
finer approximations of the first singular vectors, that involves weighted row
and column sums, we refer the reader to [6].

If the results of the tournament where only determined by the teams offense
and defense scores, then the tournament performance matrix would be

$A_{1}=\left(\begin{array}[]{cccc}1.1984&1.0459&1.4497&1.9172\\\
0.6538&0.5706&0.7909&1.0459\\\ 0.7275&0.6349&0.8800&1.1638\\\
0.9237&0.8062&1.1174&1.4777\\\ \end{array}\right)$

For example, the score of $A_{1}(2,3)=0.7909$ means that Poland should score
$0.7909$ goals against Mexico. Similarly, the score of $A_{1}(4,1)=0.9237$
means that Saudi Arabia should score $0.9237$ goals against Argentina. The
diagonal entries are of interest as well. For example, the diagonal entry
$(2,2)$ equals to $0.5706$. It indicates, based on the prediction from offense
and defense scores, how Poland would fare in offense and defense
simultaneously. Do note however, that the diagonal entry at $(i,i)$ in the
matrix $A_{1}$ is no longer the mean of the off diagonal entries in the row
$i$ and column $i$.

The amount of goal scoring during the tournament that is explained by the
teams’ offense and defense scores is

$\frac{\sigma_{1}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}+\sigma_{4}^{2}}=0.7144$

indicating that around $71$ % of goals scored during the tournament can be
explained by the teams offense and defense scores.

###  2.2 The Correction Scores

Giving interpretation to the second order singular vectors in the decompostion
should be quite interesting. We expect students to give ample feedback here as
the interpretations are not immediate. Consider the (goal) correction matrix

$A-A_{1}=\left(\begin{array}[]{cccc}-0.0317&0.9541&0.5503&-0.9172\\\
-0.6538&0.0961&-0.7909&0.9541\\\ -0.7275&-0.6349&-0.0467&0.8362\\\
1.0763&-0.8062&-0.1174&-0.1444\\\ \end{array}\right)$

with the understanding of the entries as follows. The score of $0.9541$,
Argentina vs Poland, indicates that when Argentina actaully played Poland, the
number of goals that Argentina scored against Poland was $0.9541$ higher than
the offense and defense scores would predict. Similarly, the score of
$-0.1174$, Saudi Arabia vs Mexico, indicates that when Saudi Arabia actaully
played Mexico, the number of goals that Saudi Arabia scored against Mexico was
$0.1174$ lower than the offense and defense scores would predict. The diagonal
entry at $(4,4)$ has a value of $-0.1444$. It indicates that the actual amount
of goals Saudi Arabia scored and allowed was lower than was predicted by the
offense and defense scores. We now give interpretation to the vectors

${\bf u}_{2}=(0.5771,-0.5899,-0.5547,0.1057)^{T}\mbox{ ; }{\bf
v}_{2}=(0.4186,0.3600,0.3774,-0.7435)^{T}$

We are approximating, in the least squares sense, the correction matrix
$A-A_{1}$ by a rank one matrix

$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle\sigma_{2}{\bf u}_{2}{\bf
v}^{T}_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0.5106&0.4392&0.4604&-0.9069\\\
-0.5219&-0.4489&-0.4706&0.9270\\\ -0.4907&-0.4221&-0.4425&0.8716\\\
0.0935&0.0804&0.0843&-0.1660\\\ \end{array}\right)$

The matrix $A_{2}$ can be seen as a (rank one) predicted correction matrix. It
predicts the number of goals to be scored by a team during a match as a
correction to the expected number of goals to be scored based on the offense
defense scores of the teams. The matrix $A_{2}$ is the best least squares rank
one correction matrix, noting that the actual goals scored during the game
involve two higher level correction matrices $A_{3}$, and $A_{4}$. The entries
in the vector ${\bf u}_{2}$ and ${\bf v}_{2}$ can be thought of as a
correction inducing scores, offense and defense respectively. Positive scores
reinforce, negative scores oppose. We group the positive offense correction
scores, ranked by magnitude, {Argentina, Saudi Arabia} and group the negative
offense correction scores, ranked by absolute value of the magnitude, {Poland,
Mexico}.

Similarly, we group the positive defense correction scores, ranked by
magnitude, {Argentina, Mexico, Poland} and detect the negative defense
correction score for {Saudi Arabia}. When a team $i$ in the positive offense
correction group plays a team $j$ in the positive defense correction group we
expect the team $i$ to score more goals than expected (based on the offense
defense scores) against the team $j$, proportional to magnitude of the
respective correction scores. When a team $i$ in the negative offense
correction group plays a team $j$ in the negative defense correction group we
also expect the team $i$ to score more goals than expected against the team
$j$, proportional to absolute values of the respective correction scores.
However, when a team $i$ in the positive offense correction group plays a team
$j$ in the negative defense correction group we expect the team $i$ to score
fewer goals than expected against the team $j$, proportional to absolute
values of the respective correction scores. Identical situation is detected
for the negative offense correction group and the positive defense correction
group combination.

The diagonal entries are of interest as well. The diagonal entry $(2,2)$
equals to $-0.4489$, correponds to team Poland. It indicates the team Poland
is expected to correct with fewer goals scored and allowed than the offense
defense team scores would indicate. Loosely said, Poland is predicted to have
less goal action around them than predicted based on offense defense ability.
On the other hand, the diagonal entry $(1,1)$ equals to $0.5106$, corresponds
to team Argentina. Argentina is predicted to have more goal action around them
than predicted based on offense defense ability. We can assert, on some level,
that the teams Poland, Mexico and Saudi Arabia were predicted to be more
boring than expected (based on the offense defense skills). Ranking of
predicted boredom is given by the absolute value of the diagonal entry, making
Poland the most boring predicted and Saudi Arabia the least. Argentina is
predicted to be more exciting.

We reiterate that the matrix $A_{2}$ is just the best rank one prediction for
the actual correction matrix $A-A_{1}$, based on the defense and offense
correction scores for the teams. The actual results in the tournament are
different as the matrices $A_{3}$ and $A_{4}$ were not involved. For example,
the predicted outcomes for the matches of Saudi Arabia using its offense
correction score were substantially different than the correction matrix
$A-A_{1}$ actually corrected. In particular, the match Saudi Arabia vs Mexico
was predicted by their correction scores to correct with more goals for Saudi
Arabia, when in fact the actual result in the tournament was the opposite.

As mentioned above the expected results drawn from the offense and defense
team scores and the correction offense and defense scores can be utilized for
the scenario where the teams play again in a round robin tournament under the
same attributes. The techniques discussed can be applied to any round robin
tournament of any size in any relevant sport settings.

##  3 Polar Decomposition

Polar decompostion of a matrix is a closely related result to the singular
value decomposition of a matrix, albeit its applications are not seen as
often. Having said that, the polar decomposition has a rightful place in the
context of round robin tournament matrices. We believe students of linear
algebra will find this place for the polar decomposition quite intuitive,
expecting many what if scenarios. Let us return to the results of the group C.
The teams involved had their offense attributes captured by the singular
vectors $\left\\{{\bf u}_{1},{\bf u}_{2},{\bf u}_{3},{\bf u}_{4}\right\\}$ and
their defense attributes captured by $\left\\{{\bf v}_{1},{\bf v}_{2},{\bf
v}_{3},{\bf v}_{4}\right\\}$. We now address the hypothetical tournament
outcome where the teams’ defense attributes are identical to those of their
offense attributes. In particular, ${\bf v}_{i}={\bf u}_{i}$ for all
$i\in\left\\{1,2,3,4\right\\}$. The condition ${\bf v}_{1}={\bf u}_{1}$ would
force the defense ability scores for the teams to match their offense ability
scores. This would mean that a team that has a good offense ability would in
turn suffer on defense ability and vice versa. High score on offense
translates to the same high score on defense which indicates a poor defense.
We expect lively discussions among students in regards to this, how to
interpret the forthcoming symmetric tournament matrices in the polar
decomposition. The singular value decomposition of the matrix $A$ can be
written in a polar decomposition of the matrix $A$

$A=UDV^{T}=\left(UDU^{T}\right)\left(UV^{T}\right)$

We set $P=UDU^{T}$ and the entries in the matrix $P$ provide the answer to our
hypothetical tournament scenario. The matrix $P$ is a symmetric matrix,
positive definite. We have

$P=\left(\begin{array}[]{cccc}2.9772&0.5151&0.5359&0.9721\\\
0.5151&1.7407&1.0040&0.3757\\\ 0.5359&1.0040&1.7266&0.6467\\\
0.9721&0.3757&0.6467&2.2964\\\ \end{array}\right)$

Observe $||P||_{2}=||A||_{2}$. The matrix $P$ gives the performance results
for the tournament under the hypothetical setting where the teams’ defense
attributes reflect those of their offense attributes. Each match between a
team $i$ and team $j$ would end up in a draw. For example, if Poland played
Saudi Arabia we would expect a score of $0.3757:0.3757$. Similarly, if
Argentina played Mexico we would expect a score of $0.5359:0.5359$.

Equivalently, we now address the hypothetical tournament outcome where the
teams’ offense attributes reflect those of their defense attributes. Now we
have ${\bf u}_{i}={\bf v}_{i}$ for all $i\in\left\\{1,2,3,4\right\\}$. In
particular, ${\bf u}_{1}={\bf v}_{1}$ would force the offense ability scores
for the teams to match their defense ability scores. This would mean that a
team that has a bad defense would have good offense and vice versa. We have
the following

$A=UDV^{T}=\left(UV^{T}\right)\left(VDV^{T}\right)$

We set $Q=VDV^{T}$ and the entries in the matrix $Q$ provide the answer to our
hypothetical tournament scenario. The matrix $Q$ is a symmetric matrix,
positive definite. We observe

$Q=\left(\begin{array}[]{cccc}2.0168&0.3150&0.9318&0.5711\\\
0.3150&1.8103&0.9089&0.4919\\\ 0.9318&0.9089&1.8250&0.8183\\\
0.5711&0.4919&0.8183&3.0887\\\ \end{array}\right)$

We also have $||Q||_{2}=||A||_{2}$. The matrix $Q$ gives the performance
results for the tournament under this hypothetical setting where the teams
offense attributes reflect those of their defense attributes. Once again, each
match between a team $i$ and team $j$ would end up in a draw. For example, if
Poland played Saudi Arabia in this setting we would expect a score of
$0.4919:0.4919$. Similarly, if Argentina played Mexico in this setting we
would expect a score of $0.9318:0.9318$.

Note that the diagonal entries in both $P$ and $Q$ are no longer the means of
the off diagonal entries in the corresponding rows and columns. For more on a
polar decomposition of a matrix see [4].

##  4 FIFA Groups

The draw procedure used in the $2022$ FIFA World Cup draw on the $1$st of
April $2022$ is described here. The $32$ teams were divided into four pots
based on the FIFA World Ranking announced on the $31$st of March $2022$. The
pot $1$ contained the host Qatar (assigned to group A) and the seven highest-
ranked teams. The pot $2$ contained the teams ranked $8$th to $15$th and the
pot $3$ included the teams ranked $16$th to $23$rd, while the pot $4$
contained the teams ranked $24$th to $28$th and the two placeholders from the
two inter-confederation play-offs and the winner of the UEFA play-off Path A.
The draw sequence started with the pot $1$ and ended with pot $4$. Each pot
was emptied before moving on to the next pot. Some draw conditions ensured
geographic separation. For instance, no group can have more than one team from
any continental confederation except for UEFA (AFC, CAF, CONMEBOL, CONCACAF)
and each group should consist of at least one but no more than two European
teams, see [1].

As a result the World Cup had $8$ groups of $4$ teams. The top $2$ teams from
each group advanced to the knockout stage. We now perform the singular value
decomposition for all the round robin results for all the groups.

Group A

| Netherlands | Senegal | Ecuador | Qatar
---|---|---|---|---
Netherlands | x | 2:0 | 1:1 | 2:0
Senegal | 0:2 | x | 2:1 | 3:1
Ecuador | 1:1 | 1:2 | x | 2:0
Qatar | 0:2 | 1:3 | 0:2 | x

$A=\left(\begin{array}[]{cccc}1&2.00&1.00&2.00\\\ 0.00&1.5&2.00&3.00\\\
1.00&1.00&1.6667&2.00\\\ 0.00&1.00&0.00&1.3333\\\ \end{array}\right)$

Explained by offense and defense scores: $0.9302$

${\bf u}_{1}=(0.5200,0.6534,0.4909,0.2486)^{T}\mbox{ ; }{\bf
v}_{1}=(0.1728,0.4716,0.4520,0.7372)^{T}$

${\bf u}_{2}=(0.6962,-0.5597,-0.1970,0.4039)^{T}\mbox{ ; }{\bf
v}_{2}=(0.4202,0.6395,-0.6326,-0.1197)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
Senegal, Netherlands, Ecuador, Qatar. Defense: Netherlands, Ecuador,Senegal,
Qatar. The total goal count yields the same ranking, albeit induces ties.

Group B

| England | USA | Iran | Wales
---|---|---|---|---
England | x | 0:0 | 6:2 | 3:0
USA | 0:0 | x | 1:0 | 1:1
Iran | 2:6 | 0:1 | x | 2:0
Wales | 0:3 | 1:1 | 0:2 | x

$A=\left(\begin{array}[]{cccc}1.8333&0.00&6.00&3.00\\\ 0.00&0.5&1.00&1.00\\\
2.00&0.00&1.8333&2.00\\\ 0.00&1.00&0.00&1.1667\\\ \end{array}\right)$

Explained by offense and defense scores: $0.9266$

${\bf u}_{1}=(0.8968,0.1698,0.4013,0.0761)^{T}\mbox{ ; }{\bf
v}_{1}=(0.3169,0.0209,0.8142,0.4859)^{T}$

${\bf u}_{2}=(-0.3400,0.2428,0.5151,0.7485)^{T}\mbox{ ; }{\bf
v}_{2}=(0.2382,0.5092,-0.4992,0.6593)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
England, Iran, USA, Wales . Defense: USA, England, Wales, Iran. The total goal
count yields the same ranking.

Group D

| France | Australia | Tunisia | Denmark
---|---|---|---|---
France | x | 4:1 | 0:1 | 2:1
Australia | 1:4 | x | 1:0 | 1:0
Tunisia | 1:0 | 0:1 | x | 0:0
Denmark | 1:2 | 0:1 | 0:0 | x

$A=\left(\begin{array}[]{cccc}1.5&4.00&0.00&2.00\\\ 1.00&1.1667&1.00&1.00\\\
1.00&0.00&0.3333&0.00\\\ 1.00&0.00&0.00&0.6667\\\ \end{array}\right)$

Explained by offense and defense scores: $0.8918$

${\bf u}_{1}=(0.9191,0.3619,0.0801,0.1338)^{T}\mbox{ ; }{\bf
v}_{1}=(0.3832,0.8036,0.0762,0.4489)^{T}$

${\bf u}_{2}=(0.3303,-0.5100,-0.5812,-0.5412)^{T}\mbox{ ; }{\bf
v}_{2}=(-0.7402,0.4728,-0.4581,-0.1368)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
France,Australia, Denmark, Tunisia. Defense: Tunisia, France, Denmark,
Australia. The total goal count yields the same ranking, albeit induces ties.

Group E

| Japan | Spain | Germany | Costa Rica
---|---|---|---|---
Japan | x | 2:1 | 2:1 | 0:1
Spain | 1:2 | x | 1:1 | 7:0
Germany | 1:2 | 1:1 | x | 4:2
Costa Rica | 1:0 | 0:7 | 2:4 | x

$A=\left(\begin{array}[]{cccc}1.1667&2.00&2.00&0.00\\\ 1.00&2&1.00&7.00\\\
1.00&1.00&1.8333&4.00\\\ 1.00&0.00&2.00&2.3333\\\ \end{array}\right)$

Explained by offense and defense scores: $0.8782$

${\bf u}_{1}=(0.1428,0.7958,0.4989,0.3122)^{T}\mbox{ ; }{\bf
v}_{1}=(0.1930,0.2586,0.2851,0.9026)^{T}$

${\bf u}_{2}=(0.8832,-0.3527,0.1359,0.2777)^{T}\mbox{ ; }{\bf
v}_{2}=(0.3603,0.3952,0.7323,-0.4216)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
Spain, Germany, Costa Rica, Japan. Defense: Japan,Spain, Germany, Costa Rica.
The total goal count yields different offense ranking. Offense: Spain,
Germany, Japan, Costa Rica. Defense ranking is the same.

Group F

| Morocco | Croatia | Belgium | Canada
---|---|---|---|---
Morocco | x | 0:0 | 2:0 | 2:1
Croatia | 0:0 | x | 0:0 | 4:1
Belgium | 0:2 | 0:0 | x | 1:0
Canada | 1:2 | 1:4 | 0:1 | x

$A=\left(\begin{array}[]{cccc}0.8333&0.00&2.00&2.00\\\
0.00&0.8333&0.00&4.00\\\ 0.00&0.00&0.5&1.00\\\ 1.00&1.00&0.00&1.5\\\
\end{array}\right)$

Explained by offense and defense scores: $0.8334$

${\bf u}_{1}=(0.4799,0.7784,0.2072,0.3476)^{T}\mbox{ ; }{\bf
v}_{1}=(0.1473,0.1964,0.2096,0.9465)^{T}$

${\bf u}_{2}=(0.8467,-0.4965,0.1338,-0.1369)^{T}\mbox{ ; }{\bf
v}_{2}=(0.2895,-0.2803,0.8962,-0.1854)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
Croatia, Morocco, Canada, Belgium. Defense: Morocco, Croatia, Belgium, Canada.
The total goal count yields the same ranking, albeit induces ties.

Group G

| Brazil | Switzerland | Cameroon | Serbia
---|---|---|---|---
Brazil | x | 1:0 | 0:1 | 2:0
Switzerland | 0:1 | x | 1:0 | 3:2
Cameroon | 1:0 | 0:1 | x | 3:3
Serbia | 0:2 | 2:3 | 3:3 | x

$A=\left(\begin{array}[]{cccc}0.6667&1.00&0.00&2.00\\\
0.00&1.1667&1.00&3.00\\\ 1.00&0.00&1.3333&3.00\\\ 0.00&2.00&3.00&2.1667\\\
\end{array}\right)$

Explained by offense and defense scores: $0.8565$

${\bf u}_{1}=(0.3203,0.5207,0.5007,0.6129)^{T}\mbox{ ; }{\bf
v}_{1}=(0.1135,0.3420,0.4808,0.7994)^{T}$

${\bf u}_{2}=(0.3974,0.2011,0.4696,-0.7623)^{T}\mbox{ ; }{\bf
v}_{2}=(0.3352,-0.4073,-0.6661,0.5273)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
Serbia, Switzerland, Cameroon, Brazil. Defense: Brazil, Switzerland, Cameroon,
Serbia . The total goal count yields the same ranking, albeit induces ties.

Group H

| Portugal | South Korea | Uruguay | Ghana
---|---|---|---|---
Portugal | x | 1:2 | 2:0 | 3:2
South Korea | 2:1 | x | 0:0 | 2:3
Uruguay | 0:2 | 0:0 | x | 2:0
Ghana | 2:3 | 3:2 | 0:2 | x

$A=\left(\begin{array}[]{cccc}1.1667&1.00&2.00&3.00\\\
2.00&1.3333&0.00&2.00\\\ 0.00&0.00&0.6667&2.00\\\ 2.00&3.00&0.00&2.0000\\\
\end{array}\right)$

Explained by offense and defense scores: $0.8309$

${\bf u}_{1}=(0.5752,0.4809,0.2508,0.6124)^{T}\mbox{ ; }{\bf
v}_{1}=(0.4593,0.4908,0.2118,0.7094)^{T}$

${\bf u}_{2}=(0.5954,-0.2229,0.4993,-0.5887)^{T}\mbox{ ; }{\bf
v}_{2}=(-0.3590,-0.5676,0.5892,0.4492)^{T}$

The singular value decomposition scores yield the following ranking. Offense:
Ghana, Portugal, South Korea, Uruguay . Defense: Uruguay, Portugal, South
Korea, Ghana. The total goal count yields different offense ranking. Offense:
Portugal. Ghana, South Korea, Uruguay. Defense ranking is the same.

## References

  * [1] L. Csató, Group draw with unknown qualified teams: A lesson from the 2022 FIFA World Cup draw, _International Journal of Sports Science & Coaching_, (2022)
  * [2] C. Hepler, P. Thangarajah and P. Zizler, Ranking in Professional Sports: An Application of Linear Algebra for Computer Science Students, _$21$ st Western Canadian Conference on Computing Education_, (2016).
  * [3] David James and Cynthia Botteron, Understanding Singular Vectors, _The College Mathematics Journal_ 44, No. 3, (2013).
  * [4] Peter Lancaster and Miron Tismenetsky, _The Theory of Matrices_ , Academic Press, 1985.
  * [5] Gilbert Strang, _Linear Algebra and Its Applications_ , Cengage, previously Brooks/Cole, 2006.
  * [6] P. Zizler, P. Thangarajah and M. Sobhanzadeh, On the Singular Value Decomposition and Ranking Techniques, _Computational Methods in Science and Technology(CMST)_ 26(1), (2013), 965–974.