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

Result R5567 on D6192: Complex matrix eigenvalue sequence
Subresult of R5566: Eigenvalue sequence for a diagonal complex matrix

Eigenvalue sequence for 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
(ii)	\begin{equation} A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \lambda \end{bmatrix} \end{equation}

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

Subresults

▶	R5568: Eigenvalue sequence for an identity complex matrix

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
(ii)	\begin{equation} A = \begin{bmatrix} \lambda & 0 & \cdots & 0 \\ 0 & \lambda & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \lambda \end{bmatrix} \end{equation}

This result is a particular case of R5566: Eigenvalue sequence for a diagonal complex matrix. $\square$