A problem by Jensen
William Knight · ACM SIGNUM Newsletter · 1972
Throughout, all matrices are real, S denotes a symmetric matrix, T a lower triangular matrix, and D a diagonal matrix. The following problem appeared in the May Signum Newsletter [1]. Given S, for what values of μ can T and D be found such that TDT t = (S - μI)? A sufficient condition is that μ not be an eigenvalue of any proper leading diagonal submatrix of S. A necessary and sufficient condition is this. Let m(ν) be the multiplicity of μ as an eigenvalue of the leading ν by ν diagonal submatrix of S, zero if μ is not an eigenvalue of that submatrix. The condition is that the sequence, m(ν), (including the element for S as an improper submatrix of itself) be nondecreasing. This follows from the lemma below.