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Permanent of a Matrix

Tool to calculate the permanent of a matrix. The Permanent of a a square matrix M is a value (similar to the determinant) denoted per(M).

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Tag(s) : Matrix

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Permanent of a Matrix

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Tool to calculate the permanent of a matrix. The Permanent of a a square matrix M is a value (similar to the determinant) denoted per(M).

Answers to Questions

How to calculate a matrix permanent?

The permanent of a matrix \( M = a_{i,j} \) is defined by $$ \operatorname{per}(M)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)} $$ with \( \sigma \) the elements of the symmetric group \( S_n \).

The permanent is like the determinant of a matrix, but without the signs - (minus).

For a 2x2 matrix, the calculation of the permanent is: $$ \begin{vmatrix} a & b\\c & d \end{vmatrix}=ad + bc $$

Example: \( M = \begin{vmatrix} 1 & 2\\3 & 4 \end{vmatrix}=1 \times 4 + 2 \times 3 = 10 \)

For higher size matrix like 3x3: $$ \operatorname{per}\left( \begin{vmatrix} a & b & c\\d & e & f\\g & h & i \end{vmatrix} \right) = a \operatorname{per}\left( \begin{vmatrix} e & f\\h & i \end{vmatrix} \right) + b \operatorname{per}\left( \begin{vmatrix} d & f\\g & i \end{vmatrix} \right) + c \operatorname{per}\left(\begin{vmatrix} d & e\\g & h \end{vmatrix} \right) \\ = aei+afh+bfg+bdi+cdh+ceg $$

The idea is the same for higher order matrices.

How to compute the permanent of a matrix 1x1?

For a 1x1 matrix, the permanent is the only item of the matrix.

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