it is a well know result that for two $n\times n$ matrices $A$ and $B$ the eigenvalues of the product $AB$ are equal to those of the product $BA$.
Are there similar results for a product of $m>2$ matrices? E.g. does the product of four $n\times n$ matrices $ABCD$ have the same eigenvalues as $ACBD$?
Thanks for your help!
Best Answer
Let check this on orthogonal matrices (e.g. rotations dim. $3 \times 3$).
We have for example $R_1R_1^TR_2R_2^T=I$ with $1$ as eigenvalues. If we change the order of rotations for example $R_1R_2(R_2R_1)^T=R_1R_2R_1^TR_2^T$ certainly if $R_1$ and $R_2$ are not commutative we would not obtain identity matrix and consequently the eigenvalues are different.