User:DVD206/Compound matrices

The compound matrices provide an important construction for the study of totally positive matrices. For a given matrix M the compound matrix C of order n is the matrix which entries are equal to the determinants of the n by n square submatrices of M arranged in the lexicographical order. Therefore, a matrix M is totally positive if and only if its compound matrices of all orders have positive entries.

From the Cauchy-Binet formula it follows that:



C(MN) = C(M)(C(N) $$

Since the compound matrix of a diagonal matrix is also diagonal, one can obtain the spectral decomposition of a compound matrix from the spectral decomposition of the original matrix. That is, if

M = SDS^{-1}, $$

then C

C(M) = C(S)C(D)C(S)^{-1}. $$

Exercise 2 (*): Let M be an n by n square matrix and Ck(M) it's compound matrices of order k. Express the entries of the inverse M-1 of M in terms of the entries of Cn and Cn-1.

Exercise 1 (**): Suppose, M is a square matrix that has a spectral decomposition as above. Prove that the eigenvalues of its compound matrix C(M) of order n are equal to products of all possible n-touples of the eigenvalues of the original matrix.