# Diagonal Matrix – Explanation & Examples

A diagonal matrix is a square matrix whose elements, other than the diagonal, are zero. There are certain conditions that must be met for a matrix to be called a diagonal matrix. Firstly, let’s check the formal definition of a diagonal matrix.

A square matrix in which all the elements except the principal diagonal are zero is known as a diagonal matrix.

In this article, we are going to take a close look at what makes a matrix diagonal, how to find diagonal matrices, properties of diagonal matrices, and the determinant of a diagonal matrix. Let’s start!

## What is a Diagonal Matrix?

A matrix to be classified as a diagonal matrix, it has to meet the following conditions:

• square matrix
• all elements (entries) of the matrix, other than the principal diagonal, has to be $0$

A square matrix is said to be:

• lower triangular if its elements above the principal diagonal are all $0$
• upper triangular if its elements below the principal diagonal are all $0$

Lower Triangular Matrix

Upper Triangular Matrix

diagonal matrix is a special square matrix that is BOTH upper and lower triangular since all elements, whether above or below the principal diagonal, are $0$.

## How to find Diagonal Matrix

To find, or identify, a diagonal matrix, we need to see if it is a square matrix and all the elements besides the principal diagonal (diagonal that runs from top left to bottom right) are $0$. Let’s take a look at the matrices shown below:

$A = \begin{pmatrix} 3 & 0 \\ 0 & { – 3 } \end {pmatrix}$

$B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & { – 2 } \end {pmatrix}$

$C = \begin{bmatrix} 10 & 0 \\ 0 & 12 \\ 0 & 12 \end {bmatrix}$

$D = \begin{bmatrix} { – 5 } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & { 9 } \end {bmatrix}$

Note the following observations about each of the $4$ matrices shown above:

• Matrix $A$ is a square matrix because its has the same number of rows and columns ($2 \times 2$ matrix). The principal diagonal has entries $3$ and $-3$, respectively. All other entries are $0$. Thus, this is a diagonal matrix.
• Matrix $B$ is a square matrix because its has the same number of rows and columns ($3 \times 3$ matrix). The principal diagonal has entries $1$, $5$, and $-2$, respectively. All other entries are $0$. Thus, this is a diagonal matrix.
• A false glance at Matrix $C$ will make us think that it is a diagonal matrix. But, first and foremost, it is not a square matrix. Hence, it cannot be a diagonal matrix.
• This one’s a bit tricky. First of all, it is in fact a square matrix ( $3 \times 3$ ). You will think that it is not a diagonal matrix.
But why? Is it because there’s a $0$ in the middle?

If you take a closer look at the definition of a diagonal matrix, you will see that nowhere does it say that the entries in the diagonal cannot be $0$! The condition is that the elements other than the diagonal has to be $0$. So, even if there are elements that are $0$ in the diagonal, it won’t matter. As long as the elements besides the diagonal are $0$, it will be a diagonal matrix.

Thus, Matrix $D$ is in fact a diagonal matrix!

This brings us to $2$ special types of diagonal matrices:

• Identity Matrix
• Zero Matrix

Identity Matrix

This is a square matrix in which all the entries in the principal diagonal are $1$ and all other elements are $0$. A $2 \times 2$ and a $3 \times 3$ identity matrices are shown below.

$2 \times 2$ identity matrix

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end {bmatrix}$

$3 \times 3$ identity matrix

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {bmatrix}$

Zero Matrix

A matrix in which all the elements are $0$.  A $2 \times 2$ and a $3 \times 3$ zero matrices are shown below.

$2 \times 2$ zero matrix

$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end {bmatrix}$

$3 \times 3$ zero matrix

$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end {bmatrix}$

Now, let’s look at some properties of diagonal matrices.

## Properties of Diagonal Matrices

There are several properties of diagonal matrices but for the purpose of this article, we will look at $3$ properties of diagonal matrices. Below, we take a look at the properties and their examples.

Property 1:

When $2$ diagonal matrices of the same order are added or multiplied together, the resultant matrix is another diagonal matrix with the same order.

Consider the matrices shown below:

$A = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end {bmatrix}$
$B = \begin{bmatrix} 1 & 0 \\ 0 & 5 \end {bmatrix}$

Now, we add both the $2 \times 2$ matrices and show that the resultant matrix is also diagonal.

$A + B = \begin{bmatrix} 3+1 & 0+0 \\ 0+0 & 4+5 \end {bmatrix}$

$A + B = \begin{bmatrix} 4 & 0 \\ 0 & 9 \end {bmatrix}$

Thus, we see that the resultant matrix, $A + B$, is also a diagonal matrix of the order $2 \times 2$.

Let’s check matrix multiplication with the same matrices. We multiply Matrix $A$ and Matrix $B$ and show that the resultant is also a diagonal matrix with the same order. Shown below:

$A \times B = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end {bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0 & 5 \end {bmatrix}$

$A \times B = \begin{bmatrix} {3\times1+0\times0} & 3\times 0 + 0 \times 5 \\ 0 \times 1 + 4 \times 0 & 0\times0 + 4 \times 5 \end {bmatrix}$

$A \times B = \begin{bmatrix} 3 & 0 \\ 0 & 20 \end{bmatrix}$

Thus, we see that the resultant matrix, $A \times B$, is also a diagonal matrix of the order $2 \times 2$.

Property 2:

The transpose of a diagonal matrix is the matrix itself.

If we have a matrix $A$, then we denote its transpose as $A^{T}$. Transposing a matrix means to flip its rows and columns. Let’s show that this property is true by calculating the transpose of Matrix $A$.

$A = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end {bmatrix}$

$A^{T} = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end {bmatrix}$

Interchanging the rows and columns produces the same matrix because of the entries besides the diagonal being $0$.

Property 3:

Under multiplication, diagonal matrices are commutative.

If we have $2$ matrices, $A$ and $B$, this means $AB = BA$. Let’s show this property by using the two matrices from above.

$A \times B = \begin{bmatrix} {3\times1+0\times0} & 3\times 0 + 0 \times 5 \\ 0 \times 1 + 4 \times 0 & 0\times0 + 4 \times 5 \end {bmatrix}$

$A \times B = \begin{bmatrix} 3 & 0 \\ 0 & 20 \end{bmatrix}$

Now,

$B \times A = \begin{bmatrix} {1\times3+0\times0} & 1\times 0 + 0 \times 4 \\ 0 \times 3 + 5 \times 0 & 0\times0 + 5 \times 4 \end {bmatrix}$

$B \times A = \begin{bmatrix} 3 & 0 \\ 0 & 20 \end{bmatrix}$

Thus, we have seen that $AB = BA$.

## Determinant of Diagonal Matrix

First, let’s look at the determinant of a $2\times2$ matrix.

Consider Matrix $M$ shown below:

$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

The determinant of this matrix is:

$det(M) = ad – bc$

One property of a diagonal matrix is that the determinant of a diagonal matrix is equal to the product of the elements in its principal diagonal.

Let’s see if it’s true by finding the determinant of the diagonal matrix shown below.

$N = \begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}$

$det(N) = (2\times8)-(0\times0) = 16$

This is in fact the product of the elements in its diagonal, $2\times8=16$.

#### Example 1

For the matrices shown below, identify whether they are diagonal matrix or not.

$A = \begin{bmatrix} {-2} & 0 \\ 0 & {-7} \end {bmatrix}$

$B = \begin{bmatrix} a & 0 \\ 0 & b \\ 0 & d \end {bmatrix}$

$C = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 11 \end {bmatrix}$

$D = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end {bmatrix}$

Solution

• Matrix A is a $2\times2$ matrix with the elements being 0 other than the diagonal. So, this is a diagonal matrix.
• Matrix B is a $3\times2$ matrix. It’s not square, so immediately we can say that it is not a diagonal matrix.
• Matrix C is a square matrix ($3\times3$). Also all the elements besides the diagonal are $0$. So, it is a diagonal matrix. Moreover, an entry of the diagonal is also $0$, it doesn’t matter as long as all the entries except the diagonal are zeros.
• Matrix D is a special type of diagonal matrix. It is a zero matrix. Therefore, it is a diagonal matrix.

#### Example 2

Will multiplying Matrix A and Matrix B result in a diagonal matrix?

$A = \begin{bmatrix} {-9} & 0 \\ 0 & 0 \end {bmatrix}$

$B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end {bmatrix}$

Solution

Matrix A is a diagonal matrix but Matrix B is not. So, multiplying  Matrix A and B will not result in a diagonal matrix.

#### Example 3

Find the determinant of the matrix shown below:

$B = \begin{bmatrix} 8 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & {-1} \end {bmatrix}$

Solution

Matrix B is a $3\times3$ diagonal matrix. Recall that the product of all the entries of the diagonal of a diagonal matrix is its determinant. Thus, we simply multiply and find the answer:

$det(B) = 8 \times 4 \times {-1} = – 32$

### Practice Questions

1. Identify which of the following matrices are diagonal matrices.
$J = \begin{pmatrix} 0 & 0 \\ 0 & {-5} \end{pmatrix}$
$K = \begin{pmatrix} 0 & 2 \\ 1 & {-1} \end{pmatrix}$
$L = \begin{bmatrix} -3 & 0 & 0 \\ 0 & {-5} & 0 \\ 0 & 0 & 3 \end{bmatrix}$
2. Calculate the determinant of the matrix shown below:
$T = \begin{bmatrix} -1 & 0 & 0 \\ 0 & {-1} & 0 \\ 0 & 0 & 4 \end{bmatrix}$
3. Given
$A = \begin{pmatrix} 2 & 0 \\ 0 & {-1} \end{pmatrix}$
$B = \begin{pmatrix} 1 & 0 \\ 1 & {-2} \end{pmatrix}$

Is $AB = BA$ ?

1. Matrix J is a square matrix. All the elements other than the principal diagonal are zeros. This is a diagonal matrix.
Matrix K is a square matrix but not all the elements, except the diagonal, are zero. Thus, this is not a diagonal matrix
Matrix L is a square matrix (3\times3). The elements other than the diagonal entries are zero. So, this is a diagonal matrix.
2. This is a diagonal matrix. We can find the determinant of this matrix by taking the product of all the 3 entries of the diagonal. Thus, the determinant is:
$det(T) = -1 \times -1 \times 4 = 4$
3. If two matrices are diagonal, the multiplication of those two matrices are commutative. Looking at Matrix A and B, we can see that Matrix A is diagonal but Matrix B is not. Hence, their multiplication will not be commutative.

Thus, $AB \neq BA$.