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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.
