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

Result R5560 on D5825: Complex matrix characteristic polynomial

Characteristic polynomial for a triangular complex matrix

Formulation 0

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

(i)	$A$ is a D5947: Triangular complex matrix

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}

Proofs

Proof 0

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

(i)	$A$ is a D5947: Triangular complex matrix

Let $z \in \mathbb{C}$ be a D1207: Complex number.

The matrix $z I_N - A$ is a triangular matrix with diagonal \begin{equation} z - A_{1, 1}, \quad z - A_{2, 2}, \quad \ldots, \quad z - A_{N, N} \end{equation} Hence, this result is a special case of R5072: Determinant of a triangular complex matrix. $\square$