A function on square matrices which can be given as a polynomial of the matrix entries. A special case of the "immanant" of a matrix. Used primarily in combinatorics, graph theory, and quantum field theory.
The permanent $\DeclareMathOperator\perm{perm} \perm$ is a function defined for square matrices. Given an $n\times n$ matrix $A = (a_{ij})_{i,j\in \{1,\dots,n\}}$, we define its permanent as $$ \perm(A) = \sum_{\sigma \in S_n} \prod_{i = 1}^n a_{i\sigma(i)} $$ where $S_n$ is the permutation group on $n$ objects.
Observe that the formula is very similar to the formula for the determinant of a matrix, which is given by $$ \det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i = 1}^n a_{i\sigma(i)} $$ with only the omission of the "sign" of the permutation $\sigma$ from the formula.
As an example, consider the $2\times 2$ matrix $$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ We can compute $$ \det(A) = ad - bc$$ while $$ \perm(A) = ad + bc$$
Many of the algebraic properties of the determinant has similar analogues for the permanent:
- The determinant is invariant under signed permutations of rows and columns. The permanent is invariant under all permutations of rows and columns.
- The determinant of the transpose $A^T$ is equal to the determinant of $A$. We also have $\perm(A^T) = \perm(A)$.
- For an $n\times n$ matrix $A$, we have $\det(\lambda A) = \lambda^n \det(A)$. Similarly $\perm(\lambda A) = \lambda^n \perm(A)$ (so that the functions $\det$ and $\lambda$ are both homogeneity $n$).
But the geometric properties of determinant that makes it so useful in linear algebra do not typically have analogues for the permanent.
- Given two square matrices $A,B$ of the same size, we have $\det(A B) = \det(A) \det(B)$. In general the statement "$\perm(AB) = \perm(A) \perm(B)$" is false.
- A matrix is invertible if and only if it has non-vanishing determinant. There are non-invertible matrices with non-vanishing permanent, as well as invertible matrices with vanishing permanent.
