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

Eigenvalue sequence for a diagonal complex matrix

Formulation 0

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

(i)	$\lambda_1, \lambda_2, \ldots, \lambda_N \in \mathbb{C}$ are each a D1207: Complex number
(ii)	\begin{equation} A = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \lambda_N \end{bmatrix} \end{equation}

Then $\lambda_1, \, \lambda_2, \, \ldots, \, \lambda_N$ is a D6192: Complex matrix eigenvalue sequence for $A$.

Proofs

Proof 0

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

(i)	$\lambda_1, \lambda_2, \ldots, \lambda_N \in \mathbb{C}$ are each a D1207: Complex number
(ii)	\begin{equation} A = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \lambda_N \end{bmatrix} \end{equation}

This result is a particular case of R5564: Eigenvalue sequence for an upper triangular complex matrix. $\square$