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Definition D5717
Complex matrix determinant

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D1207: Complex number $$\text{Det} A : = \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right)$$

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D1207: Complex number $$\text{Det} A : = \sum_{\pi \in S_N} \text{Sign}(\pi) A_{1, \pi(1)} A_{2, \pi(2)} A_{3, \pi(3)} \cdots A_{N - 1, \pi(N - 1)} A_{N, \pi(N)}$$
Subdefinitions
 ▶ Real matrix determinant
Children
 ▶ Complex matrix characteristic polynomial ▶ Real matrix determinant
Results
 ▶ Cofactor partition for a 2-by-2 complex square matrix ▶ Complex arithmetic expression for the determinant of a 2-by-2 complex square matrix ▶ Complex arithmetic expression for the determinant of a 3-by-3 complex square matrix ▶ Complex matrix determinant equals product of eigenvalues ▶ Complex matrix determinant is homogeneous with respect to multiplying a row or a column by a constant ▶ Complex matrix determinant zero iff some nonzero vector is mapped to zero ▶ Complex square matrix which has a zero column or a zero row has determinant zero ▶ Determinant of a complex identity matrix ▶ Determinant of a diagonal complex matrix with constant diagonal ▶ Determinant of a lower triangular complex matrix ▶ Determinant of a scaled complex matrix ▶ Determinant of an upper triangular complex matrix ▶ Determinant of reflected complex matrix