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Result R5560 on D5825: Complex matrix characteristic polynomial
Subresult of R5072: Determinant of a triangular complex matrix

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

$\begin{equation} \text{Det}(z I_N - A) = \prod_{n = 1}^N (z - A_{n, n}) \end{equation}$

Subresults

▶	R5562: Characteristic polynomial for a lower triangular complex matrix
▶	R5561: Characteristic polynomial for an upper triangular complex matrix
▶	R5563: Eigenvalue sequence for a triangular complex matrix

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.

$\square$