Diagonal Matrix

A matrix that consists of zeros as entries (or elements) outside the main diagonal is called a diagonal matrix.

Introduction

Square matrices are appeared with zeros. In a special case, a square matrix contains zero as non-diagonal elements but it contains elements only on principal diagonal. Due to having elements on leading diagonal and having zeros as non-diagonal elements, the square matrix is recognized as a diagonal matrix.

$M = [\begin{matrix} e_{1, 1} & 0 & 0 & \dots & 0 \\ 0 & e_{2, 2} & 0 & \dots & 0 \\ 0 & 0 & e_{3, 3} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & e_{m, m} \end{matrix}]$

The matrix is having elements $e_{1, 1}, e_{2, 2}, e_{3, 3}, \dots e_{m, m}$ only on principal diagonal but observe the elements on non-diagonal areas. All are zero elements at non-diagonal areas. Therefore, this type of matrix is called a diagonal matrix. The diagonal elements can be either equal or unequal elements.

It is simply expressed as $M = diag [\begin{matrix} e_{1, 1,} & e_{2, 2,} & e_{3, 3,} & \dots & e_{n, n} \end{matrix}]$

Example

$D$ is a square matrix of order $5 \times 5$ . It is having $25$ element in five rows and five columns.

$D = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & - 5 & 0 & 0 & 0 \\ 0 & 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 9 \end{matrix}]$

The matrix $D$ is having two types of elements. One type of elements are nonzero elements and remaining all are zeros. Nonzero elements ( $1, - 5, 7, 3$ and $9$ ) are placed on the leading diagonal and remaining non-diagonal elements are zeros. Therefore, the matrix $D$ is known as a diagonal matrix.