# 15.1: Definitions

An $$m\times n$$ matrix $$\mathbf{A}$$ is a rectangular array of numbers with $$m$$ rows and $$n$$ columns. The numbers $$m$$ and $$n$$ are the dimensions of $$\mathbf{A}$$. The numbers in the matrix are called its entries. The entry in row $$i$$ and column $$j$$ is called $$a_{ij}$$.

Some types of matrices have special names:

• A square matrix:$\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \\ -i & 1/2 &9 \end{pmatrix} \nonumber$ with $$m=n$$
• A rectangular matrix:$\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \end{pmatrix}\nonumber$ with $$m\neq n$$
• A column vector:$\begin{pmatrix} 3 \\ 5\\ -i \end{pmatrix}\nonumber$ with $$n=1$$
• A row vector:$\begin{pmatrix} 3 &-2 &4 \\ \end{pmatrix}\nonumber$ with $$m=1$$
• The identity matrix:$\begin{pmatrix} 1 &0 &0 \\ 0 &1 &0 \\ 0&0 &1 \end{pmatrix}\nonumber$ with $$a_{ij}=\delta_{i,j}$$, where $$\delta_{i,j}$$ is a function defined as $$\delta_{i,j}=1$$ if $$i=j$$ and $$\delta_{i,j}=0$$ if $$i\neq j$$.
• A diagonal matrix:$\begin{pmatrix} a &0 &0 \\ 0 &b &0 \\ 0&0 &c \end{pmatrix}\nonumber$ with $$a_{ij}=c_i \delta_{i,j}$$.
• An upper triangular matrix:$\begin{pmatrix} a &b &c \\ 0 &d &e \\ 0&0 &f \end{pmatrix}\nonumber$ All the entries below the main diagonal are zero.
• A lower triangular matrix:$\begin{pmatrix} a &0 &0 \\ b &c &0 \\ d&e &f \end{pmatrix}\nonumber$ All the entries above the main diagonal are zero.
• A triangular matrix is one that is either lower triangular or upper triangular.

## The Trace of a Matrix

The trace of an $$n\times n$$ square matrix $$\mathbf{A}$$ is the sum of the diagonal elements, and formally defined as $$Tr( \mathbf{A})=\sum_{i=1}^{n}a_{ii}$$.

For example,

$\mathbf{A}=\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \\ -i & 1/2 &9 \end{pmatrix}\; ; Tr(\mathbf{A})=12+3i \nonumber$

## Singular and Nonsingular Matrices

A square matrix with nonzero determinant is called nonsingular. A matrix whose determinant is zero is called singular. (Note that you cannot calculate the determinant of a non-square matrix).

## The Matrix Transpose

The matrix transpose, most commonly written $$\mathbf{A}^T$$, is the matrix obtained by exchanging $$\mathbf{A}$$’s rows and columns. It is obtained by replacing all elements $$a_{ij}$$ with $$a_{ji}$$. For example:

$\mathbf{A}=\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \end{pmatrix}\rightarrow \mathbf{A}^T=\begin{pmatrix} 3 &5\\ -2 &3i\\ 4&3 \end{pmatrix} \nonumber$