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Result R5565 on D6192: Complex matrix eigenvalue sequence
Subresult of R5563: Eigenvalue sequence for a triangular complex matrix

Eigenvalue sequence for a lower 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 D5949: Lower triangular complex matrix
(ii)	$\begin{equation} A = \begin{bmatrix} A_{1, 1} & 0 & \cdots & 0 \\ A_{2, 1} & A_{2, 2} & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ A_{N, 1} & A_{N, 2} & \cdots & A_{N, N} \end{bmatrix} \end{equation}$

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 a D5949: Lower triangular complex matrix
(ii)	$\begin{equation} A = \begin{bmatrix} A_{1, 1} & 0 & \cdots & 0 \\ A_{2, 1} & A_{2, 2} & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ A_{N, 1} & A_{N, 2} & \cdots & A_{N, N} \end{bmatrix} \end{equation}$

$\square$