📂Matrix Algebra

Diagonal Matrix

Diagonal Matrix¹

Let’s consider a matrix $A$ of size $n\times m$. The elements whose row and column numbers are the same, that is, $a_{ii} (1 \le i \le \min(n,m))$, are called the main diagonal elements. The imaginary line connecting the main diagonal elements is referred to as the main diagonal, or principal diagonal.

A matrix $A$, in which all elements except for the main diagonal elements are $0$, is called a diagonal matrix.

$$ A = [a_{ij}] = \delta_{ij} = \begin{cases} 1 & i=j \\ 0 & i \ne j \end{cases} $$

Here, $\delta$ is the Kronecker delta.

Explanation

$$ A=\begin{bmatrix} \color{red}{a_{11}} & 0 & 0 \\ 0 & \color{red}{a_{22}} & 0 \\ 0 & 0 & \color{red}{a_{33}} \end{bmatrix} \quad A=\begin{bmatrix} \color{red}{a_{11}} & 0 & 0 & 0 & 0 \\ 0 & \color{red}{a_{22}} & 0 & 0 & 0 \\ 0 & 0 & \color{red}{a_{33}} & 0 & 0 \\ 0 & 0 & 0 & \color{red}{a_{44}} & 0 \end{bmatrix} $$

As demonstrated in the example above, it is possible to define main diagonal elements and a diagonal matrix even if it’s not strictly a square matrix.

By definition, a diagonal matrix is both a lower triangular matrix and an upper triangular matrix.

Properties

Powers

Let’s consider a diagonal matrix $A = \begin{bmatrix} a_{ij}\end{bmatrix}$ of size $n\times n$. Then, the power of $A$ is as follows.

$$ A^{k}=\begin{bmatrix} (a_{11})^{k} & 0 & \cdots & 0 \\ 0 & (a_{22})^{k} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & (a_{nn})^{k} \end{bmatrix} $$

Inverse

The inverse of $A$ is as follows. In other words, the property regarding powers naturally extends even when $k$ is negative.

$$ A^{-1} = \begin{bmatrix} \dfrac{1}{a_{11}} & 0 & \cdots & 0 \\ 0 & \dfrac{1}{a_{22}} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \dfrac{1}{a_{nn}} \end{bmatrix} $$

Determinant

Considering cofactor expansion, the determinant of a diagonal matrix is the product of all the diagonal elements. For a diagonal matrix $n \times n$, its determinant is,

$$ \det [a_{ij}] = a_{11} \times \cdots \times a_{nn} $$