ThmDex – An index of mathematical definitions, results, and conjectures.
Eigenvalue sequence for an upper triangular complex matrix

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
 (i) $A$ is an D5948: Upper triangular complex matrix (ii) $$A = \begin{bmatrix} A_{1, 1} & A_{1, 2} & \cdots & A_{1, N} \\ 0 & A_{2, 2} & \vdots & A_{2, N} \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_{N, N} \end{bmatrix}$$
Then $A_{1, 1}, \, A_{2, 2}, \, \ldots, \, A_{N, N}$ 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) $A$ is an D5948: Upper triangular complex matrix (ii) $$A = \begin{bmatrix} A_{1, 1} & A_{1, 2} & \cdots & A_{1, N} \\ 0 & A_{2, 2} & \vdots & A_{2, N} \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_{N, N} \end{bmatrix}$$
This result is a particular case of R5563: Eigenvalue sequence for a triangular complex matrix. $\square$