I have a decent understanding of the determinant of a matrix in terms of its role in
- Telling you if a matrix is invertible (zero vs. nonzero)
- Expressing the product of a matrix's eigenvalues with multiplicities
- Representing the constant term in a matrix's characteristic polynomial
- Having geometrical interpretations
However, I was curious to learn in the operation known as the permanent has any interesting properties. The permanent is defined in the same way a determinant is, but all entries are added instead of alternating between positive and negative terms.
In particular, what are the significant applications of defining the permanent as a notable matrix operation? Also, are there any matrix problems whose solutions directly/indirectly depend on understanding the permanent?
Best Answer
Finding Permanent of square matrix is equivalent to finding:
(1) Number of perfect matching in the Bipartite graph (Biadjacencey Matrix).
(2) Number of cycle cover in the Directed graph (adjacency matrix).
You can check on Wiki page
(3) Number of monomial in Read-twice Formula Check this paper