Strong NP-completeness of a Matrix Similarity Problem

Consider the following problem: given an upper triangular matrix A, with rational entries and distinct diagonal elements, and a tolerance τ greater than or equal to 1, decide whether there exists a nonsingular matrix G, with condition number bounded by τ, such that G^(−1)AG is 2 × 2 block diagonal. This problem, which we shall refer to as DICHOTOMY, is an important one in the theory of invariant subspaces. It has recently been proved that DICHOTOMY is NP-complete. In this note we make some progress proving that DICHOTOMY is actually NP-complete in the strong sense. This outlines the “purely combinatorial” nature of the difficulty, which might well arise even in case of well scaled matrices with entries of small magnitude

Strong NP-completeness of a Matrix Similarity Problem

Consider the following problem: given an upper triangular matrix A, with rational entries and distinct diagonal elements, and a tolerance τ greater than or equal to 1, decide whether there exists a nonsingular matrix G, with condition number bounded by τ, such that G^(−1)AG is 2 × 2 block diagonal. This problem, which we shall refer to as DICHOTOMY, is an important one in the theory of invariant subspaces. It has recently been proved that DICHOTOMY is NP-complete. In this note we make some progress proving that DICHOTOMY is actually NP-complete in the strong sense. This outlines the “purely combinatorial” nature of the difficulty, which might well arise even in case of well scaled matrices with entries of small magnitude