Let $A_{1},A_{2},\dots,A_{m}$ be arbitrary $n\times n$ complex matrices. Prove that there are $n\times n$ unitary matrices $Q_{1},Q_{2},\dots,Q_{m}$ such that matrices $$Q_{1}^{*}A_{1}Q_{2},\ Q_{2}^{*}A_{2}Q_{3},\ \dots,\ Q_{m-1}^{*}A_{m-1}Q_{m},\ Q_{m}^{*}A_{m}Q_{1}$$ are all upper triangular.
I have observed that we can use regular Schur decomposition to construct $Q_{1},Q_{2},\dots Q_{m}$ that $$ Q_{1}^{*}A_{1}A_{2}\dots A_{m}Q_{1},Q_{2}^{*}A_{2}A_{3}\dots A_{m}A_{1}Q_{2},\dots $$ to be upper triangular, but I don't know how to proceed then.