on page 261 of this paper by J. Vermeer (http://www.math.technion.ac.il/iic/e..._pp258-283.pdf [Broken]) he writes The following assertions are equivalent. a) A is similar to a Hermitian matrix b) A is similar to a Hermitian matrix via a Hermitian, positive definite matrix c) A is similar to A* via a Hermitian, positive definite matrix anyway the proof of a)[itex]\Rightarrow[/itex]c) he writes: "There exists a V[itex]\in[/itex]Mn(ℂ) such that VAV-1 is Hermitian, i.e. VAV-1=(VAV-1)*=(V*)-1A*V*. We obtain: V*VA(V*V)-1=A* V*V is the required Hermitian and positive definite matrix." My questions is how do we know V*V is positive definite? I know it's Hermitian, i know that V*V has real eigenvalues and I know V*V is unitarily diagonalizable. I don't think that V*V is Hermitian is enough right? Does this mean that a matrix B being Hermitian is a sufficient but not necessary condition for B to be positive definite?