# 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.

## 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 × n given by

and

Multiplication: 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.

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## 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: $$e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$$