matrix

A matrix is a rectangular array of mathematical objects with elements or entries arranged in rows and columns, usually satisfying certain properties of addition and multiplication.

Most commonly, a matrix over a field $F$ is a rectangular array of elements of $F$ . The objects in the matrix are called its entries or its elements.

The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.
A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix, or $m$ -by- $n$ matrix, where $m$ and $n$ are called its dimensions.

An $m \times n$ matrix takes the form

A = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix})

where entry $a_{i j}$ is the entry of $A$ in row $i$ and column $j$ .
This can be abbreviated

A = {(\begin{matrix} a_{i j} \end{matrix})}_{1 \leq i \leq m, 1 \leq j \leq n}

Matrices of the same size can be added entrywise:

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

or, explicitly,

(\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}) + (\begin{matrix} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{22} & \dots & b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{m 1} & b_{m 2} & \dots & b_{m n} \end{matrix}) = (\begin{matrix} a_{11} + b_{11} & a_{12} + b_{12} & \dots & a_{1 n} + b_{1 n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \dots & a_{2 n} + b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} + b_{m 1} & a_{m 2} + b_{m 2} & \dots & a_{m n} + b_{m n} \end{matrix})

The product of a matrix with a scalar, i.e. scalar multiplication, is computed by multiplying each element of the matrix by the scalar:

c \cdot A = (c \cdot a_{i j})

or, explicitly,

c \cdot A = (\begin{matrix} c \cdot a_{11} & c \cdot a_{12} & \dots & c \cdot a_{1 n} \\ c \cdot a_{21} & c \cdot a_{22} & \dots & c \cdot a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ c \cdot a_{m 1} & c \cdot a_{m 2} & \dots & c \cdot a_{m n} \end{matrix})

The transpose of an $m \times n$ matrix $A$ is the $n \times m$ matrix $A^{T}$ , constructed such that the $i^{t h}$ row of $A$ is the $i^{t h}$ column of $A^{T}$ and the $j^{t h}$ column of $A$ is the $j^{t h}$ row of $A^{T}$ :

A = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}), A^{T} = (\begin{matrix} a_{11} & a_{21} & \dots & a_{n 1} \\ a_{12} & a_{22} & \dots & a_{n 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{1 m} & a_{2 m} & \dots & a_{n m} \end{matrix})

A matrix $A$ is said to be symmetric if $A^{T} = A$
A matrix $A$ is said to be skew-symmetric if $A^{T} = - A$

Matrix multiplication of two matrices corresponds to the composition of linear transformations represented by each matrix, and is only defined if the number of columns of the left matrix is equal to the number of rows of the right matrix.

If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then the matrix product $A B$ is the $m \times p$ matrix whose entries are given by the dot product of the corresponding row of $A$ and column of $B$ :

[A B]_{i j} = a_{i 1} b_{1} j + a_{21} b_{2} j + \dots + a_{i n} b_{n} j = \sum_{k = 1}^{n} a_{i k} b_{k j}

where $1 \leq i \leq m$ and $1 \leq j \leq p$ .