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

Characteristic polynomial for a complex diagonal matrix of constant diagonal

Formulation 0

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

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

Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then \begin{equation} \text{Det}(z I_N - A) = (z - c)^N \end{equation}

Formulation 1

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

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

Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then \begin{equation} \text{Det}(z I_N - A) = (z - c)^N \end{equation}

Proofs

Proof 0

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

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

Let $z \in \mathbb{C}$ be a D1207: Complex number.

This result is a particular case of R5559: Characteristic polynomial for a complex diagonal matrix. $\square$