Product with regular matrix
Task number: 4443
Show that for a Hermitian matrix \(\boldsymbol A\) and a regular matrix \(\boldsymbol R\) of the same order, the following holds:
\(\boldsymbol A\) is positive definite, if and only if \(\boldsymbol R\boldsymbol A\boldsymbol R^H\) is positive definite.
Solution 1
Without loss of generality, we need only prove one implication, because if \(\boldsymbol B=\boldsymbol R\boldsymbol A\boldsymbol R^H\), then \(\boldsymbol A=\boldsymbol S\boldsymbol B\boldsymbol S^H\) for \(\boldsymbol S = \boldsymbol R^{-1}\).
Now \(\boldsymbol x^H\boldsymbol R\boldsymbol A\boldsymbol R^H\boldsymbol x= \boldsymbol y^H\boldsymbol A \boldsymbol y> 0\) for \(\boldsymbol y= \boldsymbol R^H\boldsymbol x\).
Solution 2
Again, we prove only one implication. The decomposition \(\boldsymbol A=\boldsymbol L\boldsymbol L^H\) yields the decomposition \(\boldsymbol R\boldsymbol A\boldsymbol R^H=\boldsymbol R\boldsymbol L\boldsymbol L^H\boldsymbol R^H=(\boldsymbol R\boldsymbol L)(\boldsymbol R\boldsymbol L)^H\).
