Matrix Classes That Generate All Matrices with Positive Determinant
Charles R. Johnson, Dale D. Olesky, P. van den Driessche · SIAM Journal on Matrix Analysis and Applications · 2003
New factorization results dealing mainly with P-matrices and M-matrices are presented. It is proved that any matrix in $M_n (\mathbb{R})$ with positive determinant can be written as the product of three P-matrices (compared with the classical result that five positive definite matrices may be needed). It is also proved that a matrix A with positive determinant can be stabilized via multiplication by a P-matrix if and only if A is not a diagonal matrix with all diagonal entries negative. Factorization into two P-matrices is considered and characterized for n=2. Using elementary bidiagonal factorization results, it is shown that the nonsingular M-matrices, or the nonsingular totally nonnegative matrices, generate all matrices in $M_n (\mathbb{R})$ with positive determinant. Further results on products of M-matrices and inverse M-matrices are given. (A correction to this article has been appended to the pdf file.)