Constructing invariant subspaces as kernels of commuting matrices

Abstract

Given an n n matrix A over C and an invariant subspace N, a straightforward formula constructs an n n matrix N that commutes with A and has N = kerN. For Q a matrix putting A into Jordan canonical form, J = Q􀀀1AQ, we get N = Q􀀀1M where M= ker(M) is an invariant subspace for J with M commuting with J. In the formula J = PZT􀀀1Pt, the matrices Z and T are m m and P is an n m row selection matrix. If N is a marked subspace, m = n and Z is an n n block diagonal matrix, and if N is not a marked subspace, then m > n and Z is an m m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a nite-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.