The test was first published by Nathan Mantel, a biostatistician at the National Institutes of Health, in 1967.
[1] Accounts of it can be found in advanced statistics books (e.g., Sokal & Rohlf 1995[2]).
The test is commonly used in ecology, where the data are usually estimates of the "distance" between objects such as species of organisms.
The reasoning is that if the null hypothesis of there being no relation between the two matrices is true, then permuting the rows and columns of the matrix should be equally likely to produce a larger or a smaller coefficient.
In addition to overcoming the problems arising from the statistical dependence of elements within each of the two matrices, use of the permutation test means that no reliance is being placed on assumptions about the statistical distributions of elements in the matrices.
Many statistical packages include routines for carrying out the Mantel test.