Matrix

Definition

A matrix a rectangular array of numbers. The numbers in the array are called the entries in the matrix.

Remark

A matrix with only one row is called a row vector (or matrix). Similarly, a matrix with only one column is called a column vector (or matrix).

We use capital letters to denote matrices, and lowercase letter to denote numercal quantities, e.g.,

$$A=\left[\begin{array}{ccc}2& 1& 7\\ 3& 4& 2\end{array}\right]=\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]$$

When disussing matrices, we often refer to the numerical quantities are scalars.

The entry that occusr in row $i$ and column $j$ of matrix $A$ will is denoted by $a_{ij}$, or sometimes $(A{)}_{ij}$. Thus a general $m\times n$ matrix is written as

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]$$

A general $1\times n$ row vector $\mathbf{a}$ and a general $m\times 1$ column vector $\mathbf{b}$ is written as

$$\begin{array}{rl}\mathbf{a}& =\left[\begin{array}{c}{a}_1,a_2,\dots ,a_n\end{array}\right]\\ \mathbf{b}& =\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_m\end{array}\right]\end{array}$$

A matrix $A$ of size $n\times n$ is called a square matrix of order $n$, and the diagonal entries are said to be on the main diagonal of $A$.