# What Is Diagonal Matrix in Data Structure?

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Heather Bennett

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:

• 1. Scalar Multiplication: When you multiply a scalar value by a diagonal matrix, each element on the main diagonal is multiplied by the scalar, while all other elements remain zero.
• 2.

Addition and Subtraction: Adding or subtracting two diagonal matrices is as simple as adding or subtracting their corresponding elements on the main diagonals.

• 3. Matrix Multiplication: Multiplying two diagonal matrices results in another diagonal matrix, where each element on the main diagonal is obtained by multiplying the corresponding elements from both matrices.

## 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
```

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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