Matrices

Overview

Let $m, n \in Z^{+}$ and $F$ be a field. An $m$ -by- $n$ matrix $A$ over $F$ is a rectangular array of elements of $F$ with $m$ rows and $n$ columns:

A = [\begin{matrix} A_{1, 1} & \dots & A_{1, n} \\ ⋮ & ⋮ \\ A_{m, 1} & \dots & A_{m, n} \end{matrix}] .

The notation $A_{j, k}$ denotes the entry in row $j$ and column $k$ of $A$ . That is, the first index refers to the row number and the second index refers to the column number.

If $1 \leq i \leq m$ , then $A_{i, \cdot}$ denotes the $1$ -by- $n$ matrix consisting of row $i$ of $A$ . If $1 \leq j \leq n$ , then $A_{\cdot, j}$ denotes the $m$ -by- $1$ matrix consisting of column $j$ of $A$ .

Vector Space

The set of all $m$ -by- $n$ matrices with entries in field $F$ is denoted by $F^{m, n}$ . With addition and scalar multiplication as defined below, $F^{m, n}$ is a vector space with dimension $m n$ .

Addition

Let $A$ and $B$ be $m$ -by- $n$ matrices over field $F$ :

A = [\begin{matrix} A_{1, 1} & \dots & A_{1, n} \\ ⋮ & ⋮ \\ A_{m, 1} & \dots & A_{m, n} \end{matrix}], B = [\begin{matrix} B_{1, 1} & \dots & B_{1, n} \\ ⋮ & ⋮ \\ B_{m, 1} & \dots & B_{m, n} \end{matrix}]

The sum of $A$ and $B$ , denoted $A + B$ , is the matrix obtained by adding corresponding entries in $A$ and $B$ . That is,

A + B = [\begin{matrix} A_{1, 1} + B_{1, 1} & \dots & A_{1, n} + B_{1, n} \\ ⋮ & ⋮ \\ A_{m, 1} + B_{m, 1} & \dots & A_{m, n} + B_{m, n} \end{matrix}] .

Let $V$ and $W$ be finite-dimensional vector spaces over field $F$ . If $S, T \in L (V, W)$ , then

M (S + T) = M (S) + M (T) .

Scalar Multiplication

Let $A$ be an $m$ -by- $n$ matrix over field $F$ and $λ \in F$ :

A = [\begin{matrix} A_{1, 1} & \dots & A_{1, n} \\ ⋮ & ⋮ \\ A_{m, 1} & \dots & A_{m, n} \end{matrix}]

The product of $λ$ and $A$ , denoted $λ A$ , is the matrix obtained by multiplying each entry in $A$ by $λ$ . That is,

λ A = [\begin{matrix} λ A_{1, 1} & \dots & λ A_{1, n} \\ ⋮ & ⋮ \\ λ A_{m, 1} & \dots & λ A_{m, n} \end{matrix}] .

Let $V$ and $W$ be finite-dimensional vector spaces over field $F$ . If $T \in L (V, W)$ and $λ \in F$ , then

M (λ T) = λ M (T) .

Multiplication

Let $A$ be an $m$ -by- $n$ matrix and $B$ be an $n$ -by- $p$ matrix. Then $A B$ is defined to be the $m$ -by- $p$ matrix whose entry in row $i$ , column $j$ , is given by the following equation:

(A B)_{i, j} = \sum_{k = 1}^{n} A_{i, k} B_{k, j} = A_{i, ∙} B_{∙, j}

That is, the entry in row $i$ , column $k$ , of $A B$ is computed by taking row $i$ of $A$ and column $j$ of $B$ , multiplying together corresponding entries, and then summing.

Let $U$ , $V$ and $W$ be finite-dimensional vector spaces over field $F$ . If $T \in L (U, V)$ and $S \in L (V, W)$ , then

M (S T) = M (S) M (T) .

By Rows

Let $A$ be an $m$ -by- $n$ matrix and $B$ be an $n$ -by- $p$ matrix. Then we can also identify matrix multiplication on a per-row basis like so:

(A B)_{k, ∙} = A_{k, ∙} B .

Given any $1$ -by- $n$ matrix $a$ :

a = [\begin{matrix} a_{1} & \dots & a_{n} \end{matrix}],

it follows that

a B = a_{1} B_{1, ∙} + \dots + a_{n} B_{n, ∙} .

That is, $a B$ is a linear combination of the rows of $B$ , with the scalars that multiply the columns coming from $a$ .

By Columns

Let $A$ be an $m$ -by- $n$ matrix and $B$ be an $n$ -by- $p$ matrix. Then we can also identify matrix multiplication on a per-column basis like so:

(A B)_{∙, k} = A B_{∙, k} .

Given any $n$ -by- $1$ matrix $b$ :

b = [\begin{matrix} b_{1} \\ ⋮ \\ b_{n} \end{matrix}],

it follows that

A b = b_{1} A_{∙, 1} + \dots + b_{n} A_{∙, n} .

That is, $A b$ is a linear combination of the columns of $A$ , with the scalars that multiply the columns coming from $b$ .

Linear Maps

Let $v_{1}, \dots, v_{n}$ be a basis of vector space $V$ . The matrix of $v$ with respect to this basis is the $n$ -by- $1$ matrix

M (v) = [\begin{matrix} c_{1} \\ ⋮ \\ c_{n} \end{matrix}],

where $c_{1}, \dots, c_{n}$ are the scalars such that

v = c_{1} v_{1} + \dots + c_{n} v_{n} .

Next let $w_{1}, \dots, w_{m}$ be a basis of vector space $W$ and $T \in L (V, W)$ . The matrix of $T$ with respect to these bases is the $m$ -by- $n$ matrix $M (T)$ whose entries $A_{j, k}$ are defined by

T v_{k} = A_{1, k} w_{1} + \dots + A_{m, k} w_{m} .

If the bases are not clear from the context, then $M (v, (v_{1}, \dots, v_{n}))$ and $M (T, (v_{1}, \dots, v_{n}), (w_{1}, \dots, w_{m}))$ is used.

For all $1 \leq k \leq n$ , it follows that $M (T)_{∙, k} = M (T v_{k})$ .

Isomorphism

Let $V$ and $W$ be finite-dimensional vector spaces over field $F$ . Let $v_{1}, \dots, v_{n} \in V$ and $w_{1}, \dots, w_{m} \in W$ be bases. Then $M$ is an isomorphism between $L (V, W)$ and $F^{m, n}$ .

Suppose $T \in L (V, W)$ and $v \in V$ . Then

M (T v) = M (T) M (v) .