Definiteness and product
Task number: 4563
Prove or disprove: The product of positive definite matrices is positive definite.
Solution
The product of symmetric matrices need not be symmetric, so it is sufficient to find two positive definite matrices, whose product is not symmetric, e.g. \[\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 4 \\ \end{pmatrix} \cdot \begin{pmatrix} 3 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 3 \\ \end{pmatrix} = \begin{pmatrix} 7 & 5 & 1 \\ 5 & 7 & 2 \\ 0 & 4 & 12 \\ \end{pmatrix} \] If the product was symmetric (Hermitian), then it would already be positive definite. (It follows from the simultaneous diagonalizability of normal matrices, which goes beyond the 1st year LA curriculum.)
