Products of positive definite matrices. I
Charles S. Ballantine · Pacific Journal of Mathematics · 1967
For each positive integer n, this paper gives necessary and sufficient conditions (nasc) on a 2 x 2 real matrix S (of positive determinant) that S be a product of n positive definite real (symmetric 2x2) matrices.Also, when S is the product of (real 2x2) positive definite matrices Pi, P 2 , •> P%> it is shown that P lf P 2 ," ,P n , and S must satisfy a condition which roughly speaking measures by how much (depending on S) Pu Pz, -, Pn must collectively differ from scalar matrices.For n = 1 or 2, the abovementioned nasc are known.For n = 3, the nasc on S -[e d is (besides that ad -be > 0) that (e -b) 2 > A(adbe) whenever a + d ^ 0 .