Serlo: EN: Matrix multiplication

{{#invoke:Mathe für Nicht-Freaks/Seite|oben}} In this article, you learn how to multiply matrices. We will see that matrix multiplication is equivalent to the composition of linear maps. We will also prove some properties of the matrix multiplication.

Introduction

How can we multiply matrices?

In the article on matrices of linear maps, we learned how we can use matrices to describe linear maps $f:V\to W$ between finite-dimensional vector spaces $V$ and $W$. This requires fixing a basis $B$ of $V$ and a basis $C$ of $W$, with respect to which we can define the mapping matrix $M_C^B(f)$. In a plane of coordinates, this matrix descibes what the linear mapping $f$ does with a vector $v\in V$: Vorlage:Einrücken where $k_B:V\to K^n$ is the coordinate mapping with respect to $B$, which maps a vector $v={\lambda }_1\cdot b_1+\dots +{\lambda }_n\cdot b_n$ to the coordinate vector $({\lambda }_1,\dots ,{\lambda }_n{)}^T$ with respect to $B$. Similarly, $k_C:W\to K^m$ is the coordinate mapping with respect to $C$.

We can concatenate linear maps $f:V\to W$ and $g:W\to X$ by executing them one after the other, which results in a linear map $g\circ f:V\to X$. Can we define a suitable "concatenation" of matrices? By suitable, we mean that the "concatenation" of the matrices corresponding to $f$ and $g$ should become the matrix of the map $g\circ f$. We will call this "concatenation" of the matrices also the matrix product since it will turn out to behave almost like a product of numbers.

For example, let's consider two matrices $A\in K^{m\times l}$ and $B\in K^{p\times n}$ with the corresponding linear mas Vorlage:Einrücken and Vorlage:Einrücken given by matrix-vector-multiplication. Then $A$ is the matrix of $f_A$ (with respect to the standard bases in $K^l$ and $K^m$), and $B$ is the matrix of $f_B$ (with respect to the standard bases in $K^p$ and $K^n$). The product $A\cdot B$ of $A$ and $B$ should then be the matrix of $f_A\circ f_B$.

However, in order to be able to execute the maps $f_A$ and $f_B$ one after the other, the target space of $f_B$ must be equal to the space on which $f_A$ is defined. This means that $K^p=K^l$, i.e. $p=l$. Therefore, the number of columns of $A$ must be equal to the number of rows of $B$, otherwise we cannot define the product matrix $A\cdot B$.

Computing the product matrix

What is the product $A\cdot B$ of $A$ and $B$ corresponding to the map $f_A\circ f_B:K^n\to K^m$? To compute it, we need to calculate the images of the standard basis vectors $e_1,\dots ,e_n\in K^n$ under the map $f_A\circ f_B$. They will form the columns of the matrix of $f_A\circ f_B$, that is, the matrix $A\cdot B$.

We denote the entries of $A$ by $a_{ij}$ and those of $B$ by $b_{ij}$, i.e. $A=(a_{ij})\in K^{m\times p}$ and $B=(b_{ij})\in K^{p\times n}$. We also denote the desired matrix of $f_A\circ f_B$ by $A\cdot B=C=(c_{ij})\in K^{m\times n}$.

For $i\in \{1,\dots ,m\}$ and $j\in \{1,\dots ,n\}$, the entry $c_{ij}$ is given by the definition of the matrix representing $f_A\circ f_B$ by the $i$-th entry of the vector $f_A(f_B(e_j))\in K^m$. We can easily calculate it using the definition of $f_A$ and $f_B$ using the definition of matrix-vector-multiplication: Vorlage:Einrücken This defines all entries of the matrix $C$ and we conclude Vorlage:Einrücken This is exactly the product $C=A\cdot B$ of the two matrices $A$ and $B$.

Definition and rule of thumb

Mathematically, we can also understand matrix multiplication as an operation (just as the multiplication of real numbers).

However, there is an important difference to the multiplication of real numbers: With matrices, we have to make sure that the dimensions of the matrices we want to multiply match. Mathe für Nicht-Freaks: Vorlage:Hinweis Mathe für Nicht-Freaks: Vorlage:Warnung

To calculate the matrix product, we use the scheme "row times column".

Rule of thumb: row times column

According to the definition, each entry in the product $A\cdot B$ is the sum of the component-wise multiplication of the elements of the $i$-th row of $A$ with the $k$-th column of $B$. This procedure can be remembered as row times column, as shown in the figure on the right.

Concrete example

Example 1

We consider the following two matrices $A\in {ℝ}^{2\times 3}$ and $B\in {ℝ}^{3\times 4}$: Vorlage:Einrücken We are looking for the matrix product $C=A\cdot B\in {ℝ}^{2\times 4}$. This matrix has the form Vorlage:Einrücken We have to calculate the individual entries $c_{ik}$. We will do this here in detail for the entry $c_{23}$. The calculation of the other entries works analogously.

According to the formula Vorlage:Einrücken

This calculation can also be seen as the "multiplication" of the 2nd row of $A$ with the 3rd column of $B$. To illustrate this, we mark the entries from the sum in the matrices. We have the sum Vorlage:Einrücken These are the following entries in the matrices: Vorlage:Einrücken In this way, we can also determine the other entries of $C$ and obtain Vorlage:Einrücken

Example 2

We consider the following matrices $A\in {ℝ}^{1\times 3}$ and $B\in {ℝ}^{3\times 1}$: Vorlage:Einrücken In this case, we can calculate both $A\cdot B$ and $B\cdot A$. Let $C:=A\cdot B$. Then $C$ is a $1\times 1$-matrix $C=(c_{11})$. We calculate its only entry: Vorlage:Einrücken Thus, $A\cdot B=(1)$.

Let $D:=B\cdot A$. Then $D$ is a $3\times 3$-matrix. We can calculate the entries of $D$ by the scheme "row times column". For example, the first entry of $D$ is the first row of $B$ times the first column of $A$, i.e. $1\cdot 5=5$. If we do this with each entry, we get Vorlage:Einrücken

Example 3

In this example, we want to illustrate that the matrix multiplication really corresponds to "concatenating two matrices". That means, if we have two matrices $A$ and $B$ that we apply to a vector $v$, then we always have $(A\cdot B)v=A(Bv)$. As an example, let $A\in {ℂ}^{2\times 2}$ and $B\in {ℂ}^{2\times 3}$ be the following matrices with entries in $ℂ$: Vorlage:Einrücken Let further $v=(2,0,1{)}^T$. We check that $(A\cdot B)v=A(Bv)$. To do so, we first calculate the matrix product $A\cdot B$: Vorlage:Einrücken Now we multiply this matrix with $v$: Vorlage:Einrücken Next, we compute $A(Bv)$. Vorlage:Einrücken We now apply $A$ to this vector: Vorlage:Einrücken Indeed, here we have $(A\cdot B)v=A(Bv)$.

Properties of matrix multiplication

We now collect a few properties of the matrix multiplication.

Shortening rule for matrices representing linear maps

The following theorem shows that matrix multiplication actually reflects the composition of linear mappings. Mathe für Nicht-Freaks: Vorlage:Satz

Distributivity of matrix multiplication

Here we must be careful that the sizes of the matrices are compatible.

Left and right neutral element of matrix multiplication

We denote the entries of the unit matrix with ${\delta }_{ij}$, i.e. $I_m=({\delta }_{ij})$. Then Vorlage:Einrücken

Further reading

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