Real matrix homotopies based on Householder reflections (Q2576240)

Let \(A\) be a real \(n\times n\) matrix with positive determinant. The author constructs mappings \(H : {\mathbb R}\to{\mathbb R}^{n\times n}\) with the following properties: \(H(0)=I_n,\,H(1)=A\); the entries of \(H(t)\) are continuous rational functions of \(t\); \(\det H(t)\) is positive for every \(t\in[0,1]\); \(t\mapsto\det H(t)\) is unimodal on \([0,1]\). If \(A=LU\), where \(U\) is an upper triangular matrix with positive diagonal entries and \(L\) is a unit lower triangular matrix, then \(H(t)\;:=[I_n+t(L-I_n)][I_n+t(U-I_n)]\) satisfies these conditions. The author also constructs several different homotheties between \(A\) and \(I_n\), each starting with a different matrix factorization of \(A\), using Householder reflections in the process.