Lets say we have a symetric matrix $A=QDQ^T$
Can anyone help me take a look at what is wrong with this argument?
$$A=QDQ^T = DQQ^T =D$$
Note: the first equality is due commutativity of any matrix with diagnal matrices. The second equality is due to QQ' = I.
The only diagonal matrices which commute with any matrices are those of the form $\lambda I$. For example, consider the matrices:
$A = [[1,0],[2,0]]$ (This matrix scales the $y$ axis by $2$ and the $x$-axis by $1$.)
$B$ the matrix that rotates the plane counter-clockwise by $90$ degrees.
If you follow where the vector $(1,0)$ goes, you will see that these two matrices / linear transformations do not commute with each other.