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

Result R5559 on D5825: Complex matrix characteristic polynomial
Subresult of R5561: Characteristic polynomial for an upper triangular complex matrix

Characteristic polynomial for a complex diagonal matrix

Formulation 0

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

(i)	$c_1, \ldots, c_N \in \mathbb{C}$ are each a D1207: Complex number
(ii)	\begin{equation} A = \begin{bmatrix} c_1 & 0 & \cdots & 0 \\ 0 & c_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & c_N \end{bmatrix} \end{equation}

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

Then \begin{equation} \text{Det}(z I_N - A) = \prod_{n = 1}^N (z - c_n) \end{equation}

Subresults

▶	R5558: Characteristic polynomial for a complex diagonal matrix of constant diagonal

Proofs

Proof 0

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

(i)	$c_1, \ldots, c_N \in \mathbb{C}$ are each a D1207: Complex number
(ii)	\begin{equation} A = \begin{bmatrix} c_1 & 0 & \cdots & 0 \\ 0 & c_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & c_N \end{bmatrix} \end{equation}

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