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=ker⁡N. 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 M=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 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.