Products of hermitian matrices and symmetries
Heydar Radjavi · Proceedings of the American Mathematical Society · 1969
It is the main purpose of this note to prove that every complex matrix with real determinant is the product of four hermitian matrices; Theorem 2 is an actually stronger result. Every real square matrix is the product of two real hermitian matrices [1]; this is a special case of our Theorem 1 which is of interest in itself, if it is indeed new. Theorem 3 was motivated by a theorem of Halmos and Kakutani [3 ] who proved that every unitary operator on an infinitedimensional Hilbert space is the product of four symmetries (i.e., operators that are hermitian and unitary). We also show that the number of factors in these results cannot be reduced in general.