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BrainHurts
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on page 261 of this paper by J. Vermeer (http://www.math.technion.ac.il/iic/e..._pp258-283.pdf ) 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?
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?
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