ThmDex – An index of mathematical definitions, results, and conjectures.

Determinant of a diagonal complex matrix with constant diagonal

Formulation 0

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that

(i)	$\lambda \in \mathbb{C}$ is a D1207: Complex number
(i)	\begin{equation} A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda \end{bmatrix} \end{equation}

Then \begin{equation} \text{Det} A = \lambda^N \end{equation}

Formulation 1

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that

(i)	$\lambda \in \mathbb{C}$ is a D1207: Complex number
(ii)	$I_N \in \mathbb{C}^{N \times N}$ is a D5699: Complex identity matrix
(iii)	\begin{equation} A = \lambda I_N \end{equation}

Then \begin{equation} \text{Det} A = \lambda^N \end{equation}

Proofs

Proof 0

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that

(i)	$\lambda \in \mathbb{C}$ is a D1207: Complex number
(i)	\begin{equation} A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda \end{bmatrix} \end{equation}

This result is a particular case of R5069: Determinant of a diagonal complex matrix. $\square$