When S and T are symmetric positive definite, ST might not even be symmetric. But its eigenvalues are still positive. Start from STx = AX and take dot products with Tx. Then prove A > 0.

Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
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Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 75E
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When S and T are symmetric positive definite, ST might not even be symmetric. But its eigenvalues are still positive. Start from STx = AX and take dot products with Tx. Then prove A > 0.

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