A diagonal matrix is a square matrix whose elements, various other than the diagonal, are zero. There are specific problems that should be met for a matrix to be referred to as a diagonal matrix. Firstly, let’s check the formal definition of a diagonal matrix.

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A square matrix in which all the facets other than the major diagonal are zero is recognized as a diagonal matrix.

In this write-up, we are going to take a cshed look at what provides a matrix diagonal, exactly how to find diagonal matrices, properties of diagonal matrices, and also 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 fulfill the complying with conditions:

square matrixall facets (entries) of the matrix, other than the principal diagonal, hregarding be \$ 0 \$

A square matrix is said to be:

lower triangular if its aspects over the major diagonal are all \$ 0 \$top triangular if its facets listed below the major diagonal are all \$ 0 \$

Lower Triangular Matrix diagonal matrix is a unique square matrix that is BOTH upper and lower triangular considering that all aspects, whether above or listed below the primary diagonal, are \$ 0 \$.

## How to uncover Diagonal Matrix

To uncover, or recognize, a diagonal matrix, we must watch if it is a square matrix and all the elements besides the primary diagonal (diagonal that runs from optimal left to bottom right) are \$ 0 \$. Let’s take a look at the matrices presented below:

\$ A = eginpmatrix 3 & 0 \ 0 & – 3 finish pmatrix \$

\$ B = eginpmatrix 1 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & – 2 end pmatrix \$

\$ C = eginbmatrix 10 & 0 \ 0 & 12 \ 0 & 12 end bmatrix \$

\$ D = eginbmatrix – 5 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 9 finish bmatrix \$

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

Matrix \$ A \$ is a square matrix bereason its has actually the exact same variety of rows and also columns (\$ 2 imes 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 very same number of rows and columns (\$ 3 imes 3 \$ matrix). The primary diagonal has entries \$ 1 \$, \$ 5 \$, and \$ -2 \$, respectively. All various other entries are \$ 0 \$. Hence, this is a diagonal matrix.A false glance at Matrix \$ C \$ will certainly make us think that it is a diagonal matrix. But, first and also foremany, it is not a square matrix. Hence, it cannot be a diagonal matrix.This one’s a little bit tricky. First of all, it is in truth a square matrix ( \$ 3 imes 3 \$ ). You will certainly think that it is not a diagonal matrix.But why? Is it bereason there’s a \$ 0 \$ in the middle?

If you take a closer look at the interpretation of a diagonal matrix, you will view that nowright here does it say that the entries in the diagonal cannot be \$ 0 \$! The condition is that the aspects various other than the diagonal hregarding be \$ 0 \$. So, also if tbelow are elements that are \$ 0 \$ in the diagonal, it won’t issue. As lengthy as the elements besides the diagonal are \$ 0 \$, it will certainly be a diagonal matrix.

Hence, Matrix \$ D \$ is in reality a diagonal matrix!

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

Identity MatrixZero Matrix

Identity Matrix

This is a square matrix in which all the entries in the major diagonal are \$ 1 \$ and also all other facets are \$ 0 \$. A \$ 2 imes 2 \$ and a \$ 3 imes 3 \$ identification matrices are shown below.

\$ 2 imes 2 \$ identity matrix

\$ eginbmatrix 1 & 0 \ 0 & 1 finish bmatrix \$

\$ 3 imes 3 \$ identification matrix

\$ eginbmatrix 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end bmatrix \$

Zero Matrix

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

\$ 2 imes 2 \$ zero matrix

\$ eginbmatrix 0 & 0 \ 0 & 0 end bmatrix \$

\$ 3 imes 3 \$ zero matrix

\$ eginbmatrix 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 finish bmatrix \$

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

## Properties of Diagonal Matrices

Tright here are numerous properties of diagonal matrices but for the objective 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 very same order are included or multiplied together, the resultant matrix is another diagonal matrix through the exact same order.

Consider the matrices shown below:

\$ A = eginbmatrix 3 & 0 \ 0 & 4 finish bmatrix \$\$ B = eginbmatrix 1 & 0 \ 0 & 5 finish bmatrix \$

Now, we add both the \$ 2 imes 2\$ matrices and also present that the resultant matrix is likewise diagonal.

\$ A + B = eginbmatrix 3+1 & 0+0 \ 0+0 & 4+5 finish bmatrix \$

\$ A + B = eginbmatrix 4 & 0 \ 0 & 9 finish bmatrix \$

Hence, we check out that the resultant matrix, \$A + B\$, is additionally a diagonal matrix of the order \$2 imes 2\$.

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

\$ A imes B = eginbmatrix 3 & 0 \ 0 & 4 finish bmatrix imes eginbmatrix 1 & 0 \ 0 & 5 end bmatrix \$

\$ A imes B = eginbmatrix 3 imes1+0 imes0 & 3 imes 0 + 0 imes 5 \ 0 imes 1 + 4 imes 0 & 0 imes0 + 4 imes 5 finish bmatrix \$

\$ A imes B = eginbmatrix 3 & 0 \ 0 & 20 endbmatrix \$

Therefore, we see that the resultant matrix, \$A imes B\$, is likewise a diagonal matrix of the order \$2 imes 2\$.

To learn even more about how we did matrix multiplication, please have actually a look at the write-up below.

Property 2:

The transpose of a diagonal matrix is the matrix itself.

If we have a matrix \$A\$, then we signify its transpose as \$A^T\$. Transposing a matrix implies to flip its rows and columns. Let’s present that this residential property is true by calculating the transpose of Matrix \$A\$.

\$ A = eginbmatrix 3 & 0 \ 0 & 4 end bmatrix \$

\$ A^T = eginbmatrix 3 & 0 \ 0 & 4 end bmatrix \$

Intertransforming the rows and columns produces the very 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 also \$B\$, this implies \$AB = BA\$. Let’s show this building by using the two matrices from over.

\$ A imes B = eginbmatrix 3 imes1+0 imes0 & 3 imes 0 + 0 imes 5 \ 0 imes 1 + 4 imes 0 & 0 imes0 + 4 imes 5 end bmatrix \$

\$ A imes B = eginbmatrix 3 & 0 \ 0 & 20 endbmatrix \$

Now,

\$ B imes A = eginbmatrix 1 imes3+0 imes0 & 1 imes 0 + 0 imes 4 \ 0 imes 3 + 5 imes 0 & 0 imes0 + 5 imes 4 end bmatrix \$

\$ B imes A = eginbmatrix 3 & 0 \ 0 & 20 endbmatrix \$

Therefore, we have actually seen that \$AB = BA\$.

## Determinant of Diagonal Matrix

First, let’s look at the determinant of a \$2 imes2\$ matrix.

Consider Matrix \$M\$ presented below:

\$ M = eginpmatrix a & b \ c & d endpmatrix \$

The determinant of this matrix is:

\$ det(M) = ad – bc \$

One home of a diagonal matrix is that the determinant of a diagonal matrix is equal to the product of the aspects in its major diagonal.

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

\$ N = eginpmatrix 2 & 0 \ 0 & 8 endpmatrix \$

\$ det(N) = (2 imes8)-(0 imes0) = 16 \$

This is in reality the product of the elements in its diagonal, \$2 imes8=16\$.

Example 1

For the matrices displayed listed below, recognize whether they are diagonal matrix or not.

\$ A = eginbmatrix -2 & 0 \ 0 & -7 end bmatrix \$

\$ B = eginbmatrix a & 0 \ 0 & b \ 0 & d end bmatrix \$

\$ C = eginbmatrix 3 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 11 finish bmatrix \$

\$ D = eginbmatrix 0 & 0 \ 0 & 0 end bmatrix \$

Solution

Matrix A is a \$2 imes2\$ matrix via the elements being 0 other than the diagonal. So, this is a diagonal matrix.Matrix B is a \$3 imes2\$ matrix. It’s not square, so automatically we can say that it is not a diagonal matrix.Matrix C is a square matrix (\$3 imes3\$). Also all the aspects besides the diagonal are \$0\$. So, it is a diagonal matrix. Furthermore, an enattempt of the diagonal is likewise \$0\$, it doesn’t matter as lengthy as all the entries except the diagonal are zeros.Matrix D is a one-of-a-kind kind of diagonal matrix. It is a zero matrix. Thus, it is a diagonal matrix.Example 2

Will multiplying Matrix A and also Matrix B cause a diagonal matrix?

\$ A = eginbmatrix -9 & 0 \ 0 & 0 finish bmatrix \$

\$ B = eginbmatrix 1 & 0 \ 1 & 1 finish bmatrix \$

Solution

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

Example 3

Find the determinant of the matrix displayed below:

\$ B = eginbmatrix 8 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & -1 end bmatrix \$

Solution

Matrix B is a \$3 imes3\$ diagonal matrix. Respeak to that the product of all the entries of the diagonal of a diagonal matrix is its determinant. Therefore, we sindicate multiply and discover the answer:

\$ det(B) = 8 imes 4 imes -1 = – 32\$

### Practice Questions

Identify which of the complying with matrices are diagonal matrices.\$ J = eginpmatrix 0 & 0 \ 0 & -5 endpmatrix \$\$ K = eginpmatrix 0 & 2 \ 1 & -1 endpmatrix \$\$ L = eginbmatrix -3 & 0 & 0 \ 0 & -5 & 0 \ 0 & 0 & 3 endbmatrix \$Calculate the determinant of the matrix displayed below:\$ T = eginbmatrix -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 4 endbmatrix \$Given\$ A = eginpmatrix 2 & 0 \ 0 & -1 endpmatrix \$\$ B = eginpmatrix 1 & 0 \ 1 & -2 endpmatrix \$

Is \$AB = BA\$ ?