Abstract :
If n and k are positive integers such that k < n, and A = [aij] is an n × n complex matrix, then let A[k] denote the k × k principal submatrix [aij]ki, J=1, and let A(k) denote the corresponding complementary principal submatrix. The inequality of Fischer guarantees us that if A set membership, variant imagen, the n × n positive semidefinite Hermitian matrices, then det A less-than-or-equals, slant det A[k]det A(k) for k, 1 less-than-or-equals, slant k < n. We present a careful analysis of the Fischer error function imagek(A) = det A[k] det A(k) — det A that is based on certain geometric invariants. This analysis leads to various expansion formulas and a matrix theoretic refinement and extension of the Fischer inequality. In particular, for each k such that 1 less-than-or-equals, slant k < n we present k + 1 matrix functions ψ0, k, ψ1, k, …, ψk, k, such that if A set membership, variant imagen, then ψ0, k(A) = det A, ψk, k(A) = det A[k]det A(k), and ψi, k(A) less-than-or-equals, slant ψi+1, k(A) for each i such that 0 less-than-or-equals, slant i less-than-or-equals, slant k − 1. As we demonstrate, the matrix functions ψ0, k, ψ1, k, …, ψk, k are closely related to the various Laplace expansion formulas for the determinant function. For example, the inequality ψ0, k(A) less-than-or-equals, slant ψ1, k(A), A set membership, variant imagen, translates into det A less-than-or-equals, slant (1/k)∑ki, J=1(−1)i+jaijdet A(ij), where A = [aij] and A(ij) denotes A with row i and column j deleted. As a bonus we obtain inequalities for the higher order differences of the sequences ψ0, k(A), ψ1, k(A), …, ψk, k(A) where A set membership, variant imagen. Specifically, we define the difference operator backward difference according to backward differencexi = xi − xi+1, and show that if A set membership, variant imagen and q + t less-than-or-equals, slant k, then (−1)q backward differenceqψt, k(A) greater-or-equal, slanted 0.
