# Four players, four numbers each. The whole team becomes a matrix

Source: https://www.footballdatascience.co.uk/learn/team-as-a-matrix
Published: 2026-09-27

> One player's performance is a vector. Stack several players together and you have a matrix, the grid that almost all of data science starts from. Multiply it by a set of weights and every player gets a rating.

## The football question

In [the first part](/learn/player-as-a-vector), one midfielder's match became a vector: a list of numbers in a fixed order. But a manager doesn't pick one player. **How do you hold a whole midfield's numbers at once, so you can compare them all?**

## The concept

Stack the players' vectors on top of each other. Each player becomes a row, and each statistic a column. That grid is a **matrix**.

Here are four midfielders from the same match:

| | Passes | Tackles | Shots | Chances created |
|---|---|---|---|---|
| Player 1 | 68 | 7 | 3 | 5 |
| Player 2 | 61 | 6 | 2 | 4 |
| Player 3 | 43 | 2 | 4 | 6 |
| Player 4 | 49 | 3 | 5 | 7 |

As a matrix:

$$M = \begin{bmatrix} 68 & 7 & 3 & 5 \\ 61 & 6 & 2 & 4 \\ 43 & 2 & 4 & 6 \\ 49 & 3 & 5 & 7 \end{bmatrix}$$

<div class="plain" markdown="1">
In plain football

- **Each row is one player.** Row 1 is exactly the vector from part 1, minus the distance column.
- **Each column is one statistic.** Column 2 is every player's tackles.
- **4 × 4** means 4 rows by 4 columns. Add a fifth player and it becomes 5 × 4.
</div>

Data science has its own words for the same thing: each row is an **observation** and each column a **feature**. Every spreadsheet of player stats, and every table a machine learning model trains on, is a matrix like this one.

## Reading a matrix

### One number: a player and a stat

Each entry has an address, row first, then column. \(M_{3,4}\) is row 3, column 4: Player 3's chances created, **6**.

### Down a column: the team

Average each column and you get the midfield's average player, a vector in its own right:

$$(55.25,\ 4.5,\ 3.5,\ 5.5)$$

<div class="plain" markdown="1">
In plain football

- On average this midfield made 55 passes, 4.5 tackles, 3.5 shots and 5.5 chances each.
- Compare any player's row with it to see who's above or below the group.
</div>

### Across the rows: team balance

Look at the matrix as a whole and the shape of the midfield jumps out. Players 1 and 2 pass and tackle: they're the ones sitting deeper. Players 3 and 4 pass less but shoot and create more: they're further forward. A balanced midfield has both kinds of row. A matrix full of rows like Player 4's would create plenty and win the ball back rarely.

<figure class="rank-chart">
<div role="img" aria-label="The four-by-four midfield matrix as a shaded grid. Players 1 and 2 are darkest on passes and tackles; Players 3 and 4 are darkest on shots and chances.">

</div>
<figcaption>The matrix as a picture. Each cell is shaded against the best figure in its column, so the shape of the midfield shows at a glance: two players sitting deeper, two further forward.</figcaption>
</figure>

## Multiplying by weights: a rating for every player

Suppose you want one number per player, a rating. Decide how much each statistic is worth, write those weights as a vector, and multiply the matrix by it. Here are some made-up weights that favour attacking play:

$$M \begin{bmatrix} 0.02 \\ 0.3 \\ 0.5 \\ 0.4 \end{bmatrix} = \begin{bmatrix} 6.96 \\ 5.62 \\ 5.86 \\ 7.18 \end{bmatrix}$$

<div class="plain" markdown="1">
In plain football

- **Each pass is worth 0.02, each tackle 0.3, each shot 0.5, each chance 0.4.**
- For each player, multiply every stat by its weight and add them up. Player 1: 68 × 0.02 + 7 × 0.3 + 3 × 0.5 + 5 × 0.4 = 6.96.
- **One multiplication rates the whole team at once.** Fantasy football points work exactly like this.
</div>

With these weights, **Player 4 is the best**, on 7.18. Now weight defending more heavily, with (0.05, 0.5, 0.2, 0.2), and the ratings become 8.5, 7.25, 5.15 and 6.35. **Player 1 is now the best** and Player 4 drops to third.

Same players, same match, different "best". The matrix holds the facts; the weights hold the opinion. Every player rating system makes that choice, whether it tells you or not.

<details markdown="1">
<summary><span>Show the maths<small>Notation, the transpose, and how matrix times vector works. Optional.</small></span></summary>

An \(m \times n\) matrix has *m* rows and *n* columns, with entry \(M_{i,j}\) in row *i*, column *j*. Our team is \(4 \times 4\); a squad of 25 players with 10 statistics would be \(25 \times 10\).

The **transpose** \(M^\top\) swaps rows and columns, so each column becomes a player instead.

Multiplying an \(m \times n\) matrix by a vector of length *n* gives a vector of length *m*. Each entry is one row multiplied entry by entry with the vector, then summed:

$$(M\mathbf{w})_i = \sum_{j=1}^{n} M_{i,j}\, w_j$$

The column averages are also a matrix product: \(\frac{1}{m} M^\top \mathbf{1}\), where \(\mathbf{1}\) is a vector of ones.

</details>

## Why it matters

A matrix is the starting point for almost everything in data science. With the whole team in one grid, you can:

- **Compare players**, row against row.
- **Analyse team balance**, by the shape of the rows.
- **Group similar player profiles**, by finding rows that look alike.
- **Feed machine learning models**, which take exactly this grid as their input.

## Limitations

- **The columns are on different scales.** Passes run into dozens, shots into single figures, so raw numbers let passes dominate. That's why the rating weights for passes are so small. Standardising each column first, as in [part 1](/learn/player-as-a-vector), is the usual fix.
- **One match is a tiny sample.** A real analysis would use per-90 averages over a season.
- **The weights are made up.** Any real rating needs weights justified by data, for example by how much each action is linked to winning.

## Try it yourself

Build a matrix for your own team's midfield from its last match: one row per player, the same four columns. Choose your own weights and rate them. Then change the weights and see whether your best player changes too.

## Further reading

- [Linear transformations and matrices](https://www.3blue1brown.com/lessons/linear-transformations), 3Blue1Brown. What a matrix really does, shown visually.
- [Matrices](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices), Khan Academy. Short lessons on reading, adding and multiplying matrices.
- [Matrix multiplication](https://www.statlect.com/matrix-algebra/matrix-multiplication), StatLect. The formal rules, including matrix times vector.