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If A and C are positive definite and symmetric matrices and B is a positive semi-definite matrix and symmetric. Would the Product (assuming the dimensions match) i.e T=ABC be positive semi-definite? A sort of sub question in this same line of though that might be easier to justify is would T'=ABA be positive semi-definite. If this is true, could someone point me in the direction of some justification?

Thanks,

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For the latter case, consider $$x = A^{-1}y$$ Since $A$ is positive definite, it is also invertible which means that we can find $y$ in order to represent any vector $x$ this way. Now we have $$x^T(ABA)x = y^T A^{-1} A B A A^{-1} y = y^T B y \ge 0 $$ since $B$ is positive semidefinite. This implies that $T'$ is positive semidefinite.

For the former case, I recommend reading through the answers to this question

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