For two square complex matrices $A, B$ of the same order, what is the name of the condition
$AB=BA$, $AB^*=B^*A$?
Is this condition popular? Here $A^*$ means the transpose conjugate of $A$.
For two square complex matrices $A, B$ of the same order, what is the name of the condition
$AB=BA$, $AB^*=B^*A$?
Is this condition popular? Here $A^*$ means the transpose conjugate of $A$.
If you assume that $B$ is a normal matrix i.e $B^{*}B=BB^{*}$ and $A$ and $B$ commute, then you will automatically get that $B^{*}A=AB^{*}$. It is called Fuglede-Putnam Theorem and it is even true for normal operators from $\mathcal{B}(H)$ - all bounded linear operators on $H$, where $H$ is a Hilbert spaces. In your case $H=\mathbb{C}^{n}$.
$A$ and $B$ "doubly commute," some say. You will find at least some evidence of this by searching for "doubly commuting matrices"
. Leaving out "matrices" will find many doubly commuting operators on Hilbert space, meaning operators that commute with each other and with each other's adjoints.