1.4. Special matrices

Some matrices have certain properties which makes them useful for various mathematical applications. Understanding special matrices and their properties is important for gaining a deeper insight into linear algebra and its practical applications.

1.4.1. Square matrix

Definition 1.8 (Square matrix)

A square matrix is a matrix that has the same number of rows and columns.

For example, the following matrices are square matrices

$$\begin{array}{r}\begin{array}{rlrlrl}& \left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),& & \left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),& & \left(\begin{array}{cccc}2& 1& 0& 0\\ 1& 2& 1& 0\\ 0& 1& 2& 1\\ 0& 0& 1& 2\end{array}\right).\end{array}\end{array}$$

1.4.2. Diagonal matrix

Definition 1.9 (Main diagonal)

The main diagonal of a square $n\times n$ matrix $A$ are the elements along the diagonal from the top-left element $a_{11}$ to the bottom right element $a_{nn}$.

For example the main diagonal of the following matrix

$$\begin{array}{r}A=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right),\end{array}$$

is $(1,5,9)$.

Definition 1.10 (Diagonal matrix)

A diagonal matrix is a square $n\times n$ matrix whose elements are all zero apart from those on the main diagonal, meaning $a_{ij}=0$ for all $i\ne j$, i.e.,

$$\begin{array}{r}\begin{array}{c}\left(\begin{array}{cccc}{a}_{11}& 0& \cdots & 0\\ 0& a_{22}& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & 0& a_{nn}\end{array}\right).\end{array}\end{array}$$

For example, the following matrices are diagonal

$$\begin{array}{r}\begin{array}{rlrlrl}& \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),& & \left(\begin{array}{ccc}3.7& 0& 0\\ 0& 12& 0\\ 0& 0& 2.1\end{array}\right),& & \left(\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right).\end{array}\end{array}$$

1.4.3. Zero matrix

Similarly to $0$ being a neutral element with respect to addition of real numbers, an $m\times n$ matrix of zeros plays the role of a neutral element for addition of matrices.

Definition 1.11 (Zero matrix)

A zero matrix (or null matrix) is an $m\times n$ matrix ${\mathbf{0}}_{m\times n}$ where all the entries are zero, that is $[\mathbf{0}{]}_{ij}=0$ for all $i$ and $j$

For example

$$\begin{array}{r}\begin{array}{rlrlrl}{\mathbf{0}}_{2\times 2}& =\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),& {\mathbf{0}}_{2\times 3}& =\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right),& {\mathbf{0}}_{3\times 1}& =\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right).\end{array}\end{array}$$

Theorem 1.5 (Properties of the zero matrix)

For any $m\times n$ matrix $A$ and the zero matrix is the neutral element with respect to matrix addition and multiplication, i.e.,

• $A+\mathbf{0}=\mathbf{0}+A=A$

• $A\mathbf{0}=\mathbf{0}A=\mathbf{0}$

1.4.4. The identity matrix

Definition 1.12 (The identity matrix)

The identity matrix is denoted by $I_n$ is an $n\times n$ diagonal matrix where the elements on the main diagonal are all 1, i.e.,

(1.6)$$\begin{array}{r}[I{]}_{ij}=\{\begin{array}{ll}1,& i=j,\\ 0,& i\ne j.\end{array}\end{array}$$

For example

$$\begin{array}{r}\begin{array}{rlrlrl}{I}_2& =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),& I_3& =\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),& I_4& =\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).\end{array}\end{array}$$

It is common to omit the subscript as it should be clear what the dimensions are from the context.

Theorem 1.6 (Properties of the identity matrix)

The identity matrix has the following properties:

• $I$ is the identify element with resepct to matrix multiplication, i.e., $IA=AI=A$

• The produce of an invertible square matrix $A$ and its inverse is the identify matrix, i.e., $A{A}^{-1}=A^{-1}A=I$

Example 1.8

Given the matrices $A=\left(\begin{array}{cc}1& 0\\ -2& 3\end{array}\right)$ and $B=\left(\begin{array}{cc}1& 2\\ 4& 3\\ -2& 1\end{array}\right)$ evaluate:

(i)   $IA$

Solution

$$\begin{array}{r}\begin{array}{rl}IA& =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ -2& 3\end{array}\right)=\left(\begin{array}{cc}1+0& 0+0\\ 0-2& 0+3\end{array}\right)\\ & =\left(\begin{array}{cc}1& 0\\ -2& 3\end{array}\right)\end{array}\end{array}$$

(ii)   $AI$

Solution

$$\begin{array}{r}\begin{array}{rl}AI& =\left(\begin{array}{cc}1& 0\\ -2& 3\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1+0& 0+0\\ -2+0& 0+3\end{array}\right)\\ & =\left(\begin{array}{cc}1& 0\\ -2& 3\end{array}\right)\end{array}\end{array}$$

(iii)   $IB$

Solution

$$\begin{array}{r}\begin{array}{rl}IB& =\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{cc}1& 2\\ 4& 3\\ -2& 1\end{array}\right)=\left(\begin{array}{cc}1+0+0& 2+0+0\\ 0+4+0& 0+3+0\\ 0+0-2& 0+0+1\end{array}\right)\\ & =\left(\begin{array}{cc}1& 2\\ 4& 3\\ -2& 1\end{array}\right)\end{array}\end{array}$$

1.4.5. Symmetric Matrix

Definition 1.13 (Symmetric matrix)

A symmetric matrix is a matrix is a square matrix where the elements are symmetric with respect to the main diagonal, i.e., $a_{ij}=a_{ji}$. A symmetric matrix is equal to its transpose

(1.7)$$A=A^{\mathsf{T}}.$$

For example, the following matrices are symmetric

$$\begin{array}{r}\begin{array}{rlrl}& \left(\begin{array}{cc}1& 2\\ 2& 3\end{array}\right),& & \left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 4\\ 3& 4& 1\end{array}\right).\end{array}\end{array}$$

1.4.6. Triangular matrices

Definition 1.14 (Upper triangular matrix)

An upper triangular matrix is a non-zero square matrix where the elements on and beneath the main diagonal are all zeros.

Definition 1.15 (Lower triangular matrix)

A lower triangular matrix is a non-zero square matrix where the elements on and above the main diagonal are all zeros.

For example, the following matrices are upper triangular matrices

$$\begin{array}{r}\begin{array}{rlrlr}\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),& & \left(\begin{array}{ccc}0& 1& 2\\ 0& 0& 3\\ 0& 0& 0\end{array}\right),& & \left(\begin{array}{cccc}0& 2& 4& 6\\ 0& 0& 8& 10\\ 0& 0& 0& 12\\ 0& 0& 0& 0\end{array}\right),\end{array}\end{array}$$

and the following matrixes are lower triangular matrices

$$\begin{array}{r}\begin{array}{rlr}\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 3& 5& 0\end{array}\right),& & \left(\begin{array}{cccc}0& 0& 0& 0\\ 2& 0& 0& 0\\ 3& 5& 0& 0\\ 7& 11& 13& 0\end{array}\right).\end{array}\end{array}$$