studybyyourself » Chapter 4 : Operations on matrix

The matrix size, noted $m \times n$, is the number of rows, respectively the number of columns that a matrix contains. A matrix is said to be square when $m=n$. Matrices of example II are of size $2 \times 3$.

1. Multiplication

We have seen in the previous chapter how coefficients and variables are separated from one another, which leads to a matrix multiplying a vector. This implies this multiplication to be defined.

Let $A$ be a matrix of size $\, {\color{orangered}{m}} \times {\color{steelblue}{n}}$ and $B$ be a matrix of size $\, {\color{grey}{m’}} \times {\color{darkred}{n’}}$.

$AB=C$ is possible only if ${\color{steelblue}{n}}={\color{grey}{m’}}$. The $C$ matrix, the product, is of size ${\color{orangered}{m}} \times {\color{darkred}{n’}}$.

The multiplication $A$ by $B$ is done by distributing $A$ rows on each columns of $B$ and by adding the products, namely $(c_{i,j}) = \sum_{k=0}^{n} a_{i,k}\,b_{k,j}$.

Example I

$\begin{equation} \left(\begin{array}{cc} \class{matrix-el-bckgr}{\phantom{-}4\phantom{-}} & \class{matrix-el-bckgr}{\phantom{-}8\phantom{-}} \\ -1 & -2 \\ 6 & 0 \\ \end{array} \right) \left( \begin{array}{cc} \class{matrix-el-bckgr}{ \phantom{-}0\phantom{-}} & 1 \\ \class{matrix-el-bckgr}{-7\phantom{-}} & -1 \end{array} \right) = \left(\begin{array}{cc} \class{matrix-el-bckgr}{ \phantom{-} 4 \cdot 0 + 8 \cdot -7 \phantom{-}} & 4 \cdot 1 + 8 \cdot -1 \\ -1 \cdot 0 + -2\cdot -7 & -1 \cdot 1 + -2 \cdot -1 \\ 6 \cdot 0 + 0 \cdot -7 & 6 \cdot 1 + 0 \cdot -1 \end{array} \right) = \left(\begin{array}{cc} \class{matrix-el-bckgr}{-56\phantom{-}} & -4 \\ 14 & 1 \\ 0 & 6 \end{array} \right) \end{equation}$

2. Addition

The addition of two matrices is possible only if they both have the same size. The corresponding elements are added. The subtraction is defined in a similar way.

Example II

$\begin{equation} \left( \begin{array}{} {\color{mediumblue}{9}} & {\color{tomato}{2}} & {\color{grey}{-3}} \\ {\color{darkorange}{1}} & {\color{forestgreen}{12}} & {\color{indigo}{8}} \\ \end{array} \right) + \left( \begin{array}{} {\color{mediumblue}{7}} & {\color{tomato}{-10}} & {\color{grey}{-6}} \\ {\color{darkorange}{0}} & {\color{forestgreen}{-1}} & {\color{indigo}{0}} \\ \end{array} \right) = \left( \begin{array}{} {\color{mediumblue}{16}} & {\color{tomato}{-8}} & {\color{grey}{-9}} \\ {\color{darkorange}{1}} & {\color{forestgreen}{11}} & {\color{indigo}{8}} \\ \end{array} \right) \end{equation}$

3. Mutiplication by a scalar (number)

All elements of the matrix are multiplied by the scalar.

Example III

$\begin{equation} \color{maroon}{-2} \left( \begin{array}{c c} -11 & 2 \\ -7 & -4 \\ 0 & -1 \\ \end{array} \right) = \begin{pmatrix} {\color{maroon}{-2}} \cdot-11 & {\color{maroon}{-2}} \cdot 2 \\ {\color{maroon}{-2}} \cdot -7\phantom{1} & {\color{maroon}{-2}} \cdot-4 \\ {\color{maroon}{-2}} \cdot \phantom{-}0\phantom{1} & {\color{maroon}{-2}} \cdot-1 \end{pmatrix} = \left( \begin{array}{c c} 22 & -4 \\ 14 & 8 \\ 0 & 2 \end{array} \right) \end{equation}$

4. Decomposition

A matrix multiplying a column matrix can be decomposed into an addition of multiplications. This is due to the previous rules.

Example IV

$\begin{equation} \begin{pmatrix} 2 & -5 \\ 3 & 2 \\ 4 & -1 \end{pmatrix} \begin{pmatrix} \lambda{_1} \\ \lambda{_2} \end{pmatrix} = \begin{pmatrix} 2 \lambda{_1} + -5 \lambda{_2} \\ 3 \lambda{_1} + 2 \lambda{_2} \\ 4 \lambda{_1} + -1 \lambda{_2} \end{pmatrix} = \lambda{_1} \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} + \lambda{_2} \begin{pmatrix} -5 \\ 2 \\ -1 \end{pmatrix} \end{equation}$

Recapitulation

Multiplication

$AB = C$ where $A$ a matrix of size $\,{\color{orangered}{m}} \times {\color{steelblue}{n}}$ and $B$ a matrix of size $\, {\color{grey}{m’}} \times {\color{darkred}{n’}}$. The multiplication is possible only if ${\color{steelblue}{n}}={\color{grey}{m’}}$. Matrix $C$ size is ${\color{orangered}{m}} \times {\color{darkred}{n’}}$.
$A B \ne AB$
Let $K=T$, the left multiplication by $P$ gives $PK=PT$
Let $K=T$, the right multiplication by $P$ gives $KP=TP$

Addition

$A + B = (a_{i,j} + b_{i,j})$

Multiplication by a scalar

$\lambda A= (\lambda a_{i,j})$

Distributivity

$A(B+C)=AB+AC$

Decomposition matrix multiplying a column matrix

$
\left(\begin{smallmatrix} v_{1} & \cdots & w_{1} \\ \vdots & \cdots & \vdots \\ v_n & \cdots & w_n \end{smallmatrix}\right)
\left(\begin{smallmatrix} \lambda_1 \\ \vdots \\ \lambda_n \end{smallmatrix}\right)
= \lambda_1 \left(\begin{smallmatrix} v_{1} \\ \vdots \\ v_n \end{smallmatrix}\right)
+ \cdots
+ \lambda_n \left(\begin{smallmatrix} w_{1} \\ \vdots \\ w_n \end{smallmatrix}\right)
$