Matrix

Suppose $m$ and $n$ are nonnegative integers. An $m$-by-$n$ matrix $A$ is a rectangular array of elements of $\mathbb{F}$ with $m$ rows and $n$ columns:

$$A=\left(\begin{array}{ccc}{A}_{1,1}& \dots & A_{1,n}\\ ⋮& & ⋮\\ A_{m,1}& \dots & A_{m,n}\end{array}\right)$$

The notation $A_{j,k}$ denotes the entry in row $j$, column $k$ of $A$.

• First index is row number
• Second index is column number

For example, if we have

$$A=\left(\begin{array}{ccc}8& 4& 5-3i\\ 1& 9& 7\end{array}\right)$$

Thus $A_{2,3}$ refers to the entry in the second row, third column of $A$, which means that

$$A_{2,3}=7$$

${\mathbb{F}}^{m,n}$

With the definition of matrix addition and scalar multiplication of matrices, we can define a vector space.

For $m$ and $n$ positive integers, the set of all $m$-by-$n$ matrices with entries in $\mathbb{F}$ is denoted by ${\mathbb{F}}^{m,n}$.

Dimension of ${\mathbb{F}}^{m,n}$

Suppose $m$ and $n$ are positive integers. With addition and scalar multiplication as defined above, ${\mathbb{F}}^{m,n}$ is a vector space with $\dim {\mathbb{F}}^{m,n}=mn$.

• The additive identity of ${\mathbb{F}}^{m,n}$ is the $m$-by-$n$ matrix all of whose entries are $0$.