Constructing Invariant Subspaces as Kernels of Commuting Matrices

Abstract

Given an n by n matrix A over the complex numbers and an invariant subspace L, this paper gives a straightforward formula to construct an n by n matrix N that commutes with A and has L equal to the kernel of N. For Q a matrix putting A into Jordan canonical form J = RAQ with R the inverse of Q, we get N = RM$ where the kernel of M is an invariant subspace for J with M commuting with J. In the formula M = P ZVW with V the inverse of a constructed matrix T and W the transpose of P, the matrices Z and T are m by m and P is an n by m row selection matrix. If L is a marked subspace, m = n and Z is an n by n block diagonal matrix, and if L is not a marked subspace, then m > n and Z is an m by m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a finite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A.