ThmDex – An index of mathematical definitions, results, and conjectures.
Characteristic polynomial for an upper triangular complex matrix
Formulation 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) \begin{equation} 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} \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 - A_{n, n}) \end{equation}
Subresults
R5559: Characteristic polynomial for a complex diagonal matrix
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) \begin{equation} 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} \end{equation}
Let $z \in \mathbb{C}$ be a D1207: Complex number.
This result is a particular case of R5560: Characteristic polynomial for a triangular complex matrix. $\square$