What properties must have two orthonormal matrix to commute?
What properties must have two orthonormal matrix to commute? I mean what properties must two orthonormal matrices $A$ and $B$ have for $AB =BA$?
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$\begingroup$Two normal matrices commute if and only if they are diagonalizable with respect to the same orthonormal basis. This is also equivalent to being unitarily equivalent, that is $B=UAU^*$ for some unitary $U$.
Here is a sketch of the argument. Suppose that $A,B$ are normal and that they commute.
Note: a matrix $A$ is "normal" if it commutes with its adjoint $A^*$ (conjugate transpose). For real matrices, $A$ normal means simply that $AA^t=A^tA$. A matrix $U$ is "unitary" if $U^*U=UU^*=I$; when $U$ is real, unitary is the same a orthogonal.
Any normal matrix is of the form $UDU^*$, for a unitary $U$ and diagonal $D$. This follows from Schur's Triangulation Theorem.
As mentioned in the aforementioned article, Schur's decomposition can be applied simultaneously (i.e., with the same unitary) to commuting matrices.
By the above, since $AB=BA$, we have $A=UDU^*$, $B=UEU^*$ with $D,E$ diagonal.
Conversely, if $A,B$ are diagonalizable with respect to the same orthonormal basis, we have $A=UDU^*$, $B=UEU^*$, where $D,E$ are diagonal and $U$ is the unitary doing the change of basis from said orthonormal basis to the canonical one. Thus$$ AB=UDU^*UEU^*=UDEU^*=UEDU^*=UEU^*UDU^*=BA. $$
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