Why permutation matrices are used to swap rows of an array?

Question

What are the advantages of using a permutation matrix to swap rows? Why one would create a permutation matrix and then apply a matrix multiplication, is it easier and more efficient than just swapping rows with a for loop?

Answer 1

Permutation matrices are a useful mathematical abstraction, because they allow analysis using the normal rules of matrix algebra, without having to introduce another type of operation.

In software, good implementations do not store a permutation matrix as a full matrix, they store a permutation array and they apply it directly (without a full matrix multiplication).

Depending on the sizes of the matrices and the operations and access patterns involved, it may be cheaper not to apply the permutation to the data in memory at all, but just to use it as an extra indirection. So, when you request (P * M)(i,j), where P is a permutation matrix and M is some other matrix that you are permuting, the data need not be re-arranged at all, but rather the element access operation will look up the permuted row when you access the element.

Answer 2

The first thing that comes into my mind is the issue called "spatial locality". Caching technologies assume that if a memory location is accessed, it is probable to access the nearby locations of the memory. In some programming languages, elements in rows are neighbors whereas elements in columns are neighbors in others. It depends on the implementation. I guess permutation matrices are designed to solve this problem, since optimization of matrix multiplication is one of the problems that algorithms academia mostly works on improving. Simple loop structure will not be able to make use of cache technologies to improve performance.