ThmDex – An index of mathematical definitions, results, and conjectures.
Eigenvalue sequence for a diagonal complex matrix

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) $$A = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \lambda_N \end{bmatrix}$$
Then $\lambda_1, \, \lambda_2, \, \ldots, \, \lambda_N$ is a D6192: Complex matrix eigenvalue sequence for $A$.
Subresults
 ▶ R5567: Eigenvalue sequence for a diagonal complex matrix with constant diagonal
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) $$A = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & \lambda_N \end{bmatrix}$$
This result is a particular case of R5564: Eigenvalue sequence for an upper triangular complex matrix. $\square$