# Diagonal Matrices

  

## Definition of a Diagonal Matrix

A diagonal matrix is a square matrix in which all entries off the main diagonal are zero.
These are examples of diagonal matrices.
$A = \begin{bmatrix} -6 & 0 \\ 0 & 9 \end{bmatrix}$     $B = \begin{bmatrix} 9 & 0 & 0\\ 0 & 7 & 0\\ 0 & 0 & -1 \end{bmatrix}$     $C = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -5 \end{bmatrix}$     $D = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

## Properties of the Diagonal Matrices

Some of the most important properties of the diagonal matrices are given below.

1.    The determinant of a diagonal matrix is equal to the product of all entries in the main diagonal.
2.    A diagonal matrix is a symmetric matrix.
3.    A diagonal matrix is invertible (it has an inverse) if and only if none of its entries in the main diagonal is zero.

## Addition and Multiplication of Diagonal Matrices

Let $A$ and $B$ be two diagonal matrices of the same size $n \times n$ given by $A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & 0 & 0 & ... \\ 0 & b_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & b_{nn} \end{bmatrix}$

Addition: $A + B = \begin{bmatrix} a_{11} + b_{11} & 0 & 0 & ... \\ 0 & a_{22} + b_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} + b_{nn} \end{bmatrix}$ is a diagonal matrix.

Mutliplication: $A B = \begin{bmatrix} a_{11} b_{11} & 0 & 0 & ... \\ 0 & a_{22} b_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} b_{nn} \end{bmatrix}$ is a diagonal matrix.

## Inverse of Diagonal Matrices

Let $A$ be a diagonal matrix given by $A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix}$

If none of the entries $a_{ii}$ in the main diagonal is equal to zero, then $A$ is invertible and its inverse is given by
$A^{-1} = \begin{bmatrix} \dfrac{1}{a_{11}} & 0 & 0 & ... \\ 0 & \dfrac{1}{a_{22}} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & \dfrac{1}{a_{nn}} \end{bmatrix}$

## Power of Diagonal Matrices

Let $A$ be a diagonal matrix given by $A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix}$

$A^p = \begin{bmatrix} a_{11}^p & 0 & 0 & ... \\ 0 & a_{22}^p & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn}^p \end{bmatrix}$ where $p$ is a positive integer.

## Eigenvalues and Eigenvectors of Diagonal Matrices

Let $A$ be a diagonal matrix given by $A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix}$

The eigenvalues $\lambda_i$ and corresponding eigenvectors $e_i$ are given by
$\lambda_1 = a_{11}$   ,   $e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ . \\ . \\ 0 \end{bmatrix}$

$\lambda_2 = a_{22}$   ,   $e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ . \\ . \\ 0 \end{bmatrix}$
...
etc
...
$\lambda_n = a_{nn}$   ,   $e_{n} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ . \\ . \\ 1 \end{bmatrix}$

## Examples with Solutions

Example 1
Which of the following are diagonal matrices?
a) $A = \begin{bmatrix} 0 & 0 \\ 1 & 2 \end{bmatrix}$      b) $B = \begin{bmatrix} -2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$      c) $C = \begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0\\ 5 & 0 & 0 & 4 \end{bmatrix}$      d) $D = \begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 9 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix}$

Solution
Matrices $B$ and $D$ are diagonal matrices.

Example 2
Matrices $A$ and $B$ are given by $A = \begin{bmatrix} -1 & 0 & 0\\ 0 & -\dfrac{2}{3} & 0 \\ 0 & 0 & \dfrac{1}{4} \end{bmatrix}$ , $B = \begin{bmatrix} -1 & 0 & 0\\ 0 & -6 & 0 \\ 0 & 0 & 8 \end{bmatrix}$.
Calculate $A B$

Solution
$A B = \begin{bmatrix} -1 & 0 & 0\\ 0 & -\dfrac{2}{3} & 0 \\ 0 & 0 & \dfrac{1}{4} \end{bmatrix} \begin{bmatrix} -1 & 0 & 0\\ 0 & -6 & 0 \\ 0 & 0 & 8 \end{bmatrix}$

$\quad \quad = \begin{bmatrix} (-1)(-1) & 0 & 0\\ 0 & (-\dfrac{2}{3})(-6) & 0 \\ 0 & 0 & (\dfrac{1}{4})(8) \end{bmatrix}$

$\quad \quad = \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0 \\ 0 & 0 & 2 \end{bmatrix}$

Example 3
Let matrix $A = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & - 2 \end{bmatrix}$
Calculate $A^5$.

Solution
$A^5 = \begin{bmatrix} (-1)^5 & 0 & 0 & 0 \\ 0 & (2)^5 & 0 & 0 \\ 0 & 0 & (-1)^5 & 0 \\ 0 & 0 & 0 & (-2)^5 \end{bmatrix}$
Simplify
$\quad \quad = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 32 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -32 \end{bmatrix}$

Example 4
Find the eigenvalues and eigenvectors of the matrix
$A = \begin{bmatrix} 1 & 0\\ 0 & -3 \end{bmatrix}$

Solution
Eigenvalue: $\lambda_1 = 1$   ,   Eigenvector: $e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_2 = -3$   ,   Eigenvector: $e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

Example 5
Find the eigenvalues and eigenvectors of the matrix
$A = \begin{bmatrix} -6 & 0 & 0\\ 0 & \dfrac{1}{5} & 0 \\ 0 & 0 & - \dfrac{2}{3} \end{bmatrix}$

Solution
Eigenvalue: $\lambda_1 = -6$   ,   Eigenvector: $e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_2 = \dfrac{1}{5}$   ,   Eigenvector: $e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_3 = - \dfrac{2}{3}$   ,   Eigenvector: $e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$

Example 6
Find the eigenvalues and eigenvectors of the matrix
$A = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 9 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

Solution
Eigenvalue: $\lambda_1 = 1$   ,   Eigenvector: $e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_2 = -1$   ,   Eigenvector: $e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_3 = 9$   ,   Eigenvector: $e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_4 = 1$   ,   Eigenvector: $e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$

## Questions (with solutions given below)

• Part 1
Find the real constants $a$ and $b$ such that $A^{-1} = A$ where $A$ is a matrix given by $A = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \dfrac{1}{a} & 0 & 0\\ 0 & 0 & b+1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$
• Part 2
Find the eigenvalues and eigenvectors of the matrix
$A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
• Part 3
Find the eigenvalues and eigenvectors of the matrix
$A = \begin{bmatrix} 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & -2 & 0\\ 0 & 0 & 0 & -2 \end{bmatrix}$

### Solutions to the Above Questions

• Part 1
We first find the inverse of the given matrix $A^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & a & 0 & 0\\ 0 & 0 & \dfrac{1}{b+1} & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$
In order to have $A^{-1} = A$, we need to have
$a = \dfrac{1}{a}$    (I) and $\dfrac{1}{b+1} = b + 1$    (II)
Solve the above equations. Equation (I) may be written as
$a^2 = 1$
and gives two solutions: $a = 1$ and $a = - 1$
Equation (II) may be written as
$(b+1)^2 = 1$
and gives two solutions: $b = 0$ and $b = - 2$
Hence the solutions to the given questions are the following pairs:
$a = 1$ , $b = 0$
$a = 1$ , $b = -2$
$a = - 1$ , $b = 0$
$a = - 1$ , $b = -2$

• Part 2

Eigenvalue: $\lambda_1 = 1$   ,   Eigenvector: $e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_2 = 1$   ,   Eigenvector: $e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_3 = 0$   ,   Eigenvector: $e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$

• Part 3
Eigenvalue: $\lambda_1 = 2$   ,   Eigenvector: $e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_2 = 2$   ,   Eigenvector: $e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_3 = -2$   ,   Eigenvector: $e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}$

Eigenvalue: $\lambda_4 = -2$   ,   Eigenvector: $e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$