I am trying to figure out whether the following implication is correct or not:
If $B$ is Hermitian and $A$ is Hermitian positive definite, then $A$ and $B$ commute.
Any ideas?
I am trying to figure out whether the following implication is correct or not:
If $B$ is Hermitian and $A$ is Hermitian positive definite, then $A$ and $B$ commute.
Any ideas?
By the following example, we can show that $A$ and $B$ don't necessarily commute:
Let $A=\begin{bmatrix} 1 & 0\\ 0 &2 \end{bmatrix} $
Let $ B=\begin{bmatrix} 1 &1+i \\ 1-i& 4 \end{bmatrix}$
$A$ is hermitian positive definite, and $B$ is hermitian. Note that: $AB=\begin{bmatrix} 1 &1+i \\ 2-2i& 4 \end{bmatrix}\neq BA=\begin{bmatrix} 1 &2+2i \\ 1-i & 4 \end{bmatrix}$