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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]M

V*VA(V*V)

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?

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]M

_{n}(ℂ) such that VAV^{-1}is Hermitian, i.e. VAV^{-1}=(VAV^{-1})*=(V*)^{-1}A*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?

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