I got this problem and am not sure how to start it and would really appreciate help!
Let $M$ and $N$ be (n × n)-complex matrices. Suppose that $M$ and $N$ are simultaneously unitarily similar to upper-triangular matrices. In other words, there exists a unitary matrix Q such that $Q^*MQ$ and $Q^*NQ$ are both upper-triangular matrices. Show that every eigenvalue of the difference $M N − N M$ must be zero.
Since eigenvalues are preserved under similarity transforms, it suffices to look at eigenvalues of $Q^*(MN-NM)Q$. Distributing the $Q$'s gives,
$Q^*MNQ-Q^*NMQ$, which by $Q Q^*=I$ produces,
(1) $Q^*MQQ^*NQ-Q^*NQ Q^*MQ$, which by the conditions is just the difference between two products of upper triangular matrices.
Now since the product of upper triangular matrices contain the pairwise products of their diagonal elements along the product's diagonal (eg if $d_{1M}$ is the first element of $Q^*MQ$'s diagonal, and $d_{1N}$ is the first element of $Q^*NQ$'s diagonal, then their product will have $d_{1M} d_{1N}$ as the first element on its diagonal). Since these product diagonals will be the same regardless of M-Q or Q-M product ordering, the difference in equation (1) will be a upper triangular matrix with zeros along its entire diagonal. It's easy to verify this means all its eigenvalues are zero.