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}

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}

Let $S_N$ be a D4951: Set of standard N-permutations.

The

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}

Let $S_N$ be a D4951: Set of standard N-permutations.

The

Subdefinitions

▶ | Real matrix determinant |

Children

▶ | Complex matrix characteristic polynomial |

▶ | Real matrix determinant |

Results