ThmDex – An index of mathematical definitions, results, and conjectures.
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Zermelo-Fraenkel set theory
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Matrix determinant
Definition D5717
Complex matrix determinant
Formulation 0
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 \begin{equation} \text{Det} A : = \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right) \end{equation}
Formulation 1
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 \begin{equation} \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)} \end{equation}
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