# Diagonal Matrix

## Diagonal Matrix Example

Any given square matrix where all the elements are zero except for the elements that are present diagonally is called a diagonal matrix. Let’s assume a square matrix [Aij]n x m can be called as a diagonal matrix if Aij= 0, if and only if i ≠ j. That is the Diagonal Matrix definition. There are many other matrices other than the Diagonal Matrix, such as symmetric matrix, antisymmetric, diagonal matrix, etc. In this section, you will be studying diagonal matrix definition, the properties of a diagonal matrix, sample solved problems of Diagonal Matrix.

### Properties of Diagonal Matrix

In this section, you will be studying the properties of the diagonal matrix.

Property 1: If addition or multiplication is being applied on diagonal matrices, then the matrices should be of the same order.

Example:

Hence, this is the diagonal matrix.

Property 2: When you transpose a diagonal matrix, it is just the same as the original because all the diagonal numbers are 0.

Property 3: Diagonal Matrices are commutative when multiplication is applied.

Hence, A x B = B x A

### What is the Block Diagonal Matrix?

A matrix that can be split into multiple different blocks is called a block matrix. In such matrices, the non-diagonal numbers are 0. Therefore, Aij = 0, where i ≠ j. Such matrices are called block-diagonal matrices.

Here’s an example of a block diagonal matrix:

### The inverse of a Diagonal Matrix

Let us consider a diagonal matrix

The determinants of the above matrix are

|D|   = x11x22x33

### Anti-Diagonal Matrix

If all the numbers in the matrix are 0 except for the diagonal numbers from the upper right corner to the lower-left corner, it is called an anti diagonal matrix. It is represented as:

### Sample Questions

Question 1: If A = B = then apply addition and find out if there is a diagonal in the matrix or not.

Solution:

Given,

Yes, when addition operation is applied between Matrix A and Matrix B, the resultant is diagonal in the matrix.

Question 2: If A =B =   then apply, multiplication, and find out if there is a diagonal in the matrix or not.

Solution:

Given,

Yes, when multiplication is applied between Matrix A and Matrix B, the resultant is a diagonal matrix.

Question 3: If A =  and B = show that multiplication is cumulative in diagonal matrices.

Solution:

Given,

Hence Proved.

Yes, multiplication operation is cumulative between Diagonal Matrix A and Diagonal Matrix B.

Question 4: Apply property 2 of a diagonal matrix and show that the transpose of a matrix is the same as the original.

Solution:

Given,

1. Define Diagonal Matrix with an Example

Any given square matrix where all the elements are zero except for the elements that are present diagonally is called a diagonal matrix. Let’s assume a square matrix [Aij]n x m can be called as a diagonal matrix if Aij= 0, if and only if i ≠ j. That is the Diagonal Matrix definition. There are many other matrices other than the Diagonal Matrix, such as symmetric matrix, antisymmetric, diagonal matrix, etc.

Example of a Diagonal Matrix =

2. What are the Properties of a Diagonal Matrix?

Property 1: If addition or multiplication is being applied on diagonal matrices, then the matrices should be of the same order.

Property 2: When you transpose a diagonal matrix, it is just the same as the original because all the diagonal numbers are 0.

Property 3: Diagonal Matrices are commutative when multiplication is applied.

3. Derive the Inverse of a Diagonal Matrix

Let us consider a diagonal matrix

The determinants of the above matrix are

|D|   = x11x22x33