ThmDex – An index of mathematical definitions, results, and conjectures.
Determinant of 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 \begin{equation} \text{Det} A = \prod_{n = 1}^N 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 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}
This result is a particular case of R5072: Determinant of a triangular complex matrix. $\square$