Given $B$ ∈ $R(n×n)$ and $B$ is invertible and positive semidefinite, show that $A$ is also positive semidefinite.
Show that $A$ is positive semidefinite given $(B^TAB)$ is positive semidefinite and $B$ is invertible
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linear-algebra
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0Are you aware of the reverse result? That is, if $A$ is PSD, then so is $B^TAB$? – 2017-02-12
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0Also, you've given us the problem statement, but you haven't given us any context. What have you tried so far? Do you have any thoughts on the problem? – 2017-02-12
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0I've proven it the other way by showing $B^TAB$ is symmetric. Im unsure how to start this proof however. – 2017-02-12
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0Well, for the other way: it's not enough to simply show that $B^TAB$ is **symmetric**. Typically, one proves this using the *definition* of positive semidefinite as opposed to (symmetry) + (non-negative eigenvalues). – 2017-02-12
1 Answers
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Hint: We know that $B^TAB$ is symmetric. So, consider the matrix $$ (B^{-1})^TB^TAB(B^{-1}) $$