A diagonal matrix is a special type of square matrix where all the elements outside the main diagonal are zero. The main diagonal of a matrix consists of elements that have the same row and column index. In other words, a diagonal matrix has non-zero elements only on its main diagonal, and all other elements are zero.

## Properties of Diagonal Matrix

A diagonal matrix has several interesting properties:

## Example

To understand better, let’s consider an example of a 3×3 diagonal matrix:

1 0 0
0 4 0
0 0 -2

In this example, the first element (1) corresponds to row 1 and column 1, second element (4) corresponds to row 2 and column 2, and so on. All other elements outside the main diagonal are zero.

### Scalar Multiplication Example

If we multiply this matrix by a scalar value of 3:

3 * (1 0 0)
(0 4 0)
(0 0 -2)

The resulting matrix will be:

3 0 0
0 12 0
0 0 -6

### Addition Example

Let’s consider another example where we add two diagonal matrices:

(1 0 0) + (2 0 0) = (3 0 0)
(0 4 0) + (0 -1 0) = (3 -1 0)
(0 2 -2) + (1 -1 -2) = (1 -1 -4)

### Matrix Multiplication Example

Lastly, let’s multiply two diagonal matrices:

(1 0 ) * (4 5 ) = (4 5 )
( ) * ( -2) = ( )
( ) * (-6 ) = (-6 )

The resulting matrix will also be a diagonal matrix:

4

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