Lemma 4.22 — simultaneous biunitary diagonalization

For every natural number $d$ with $0<d$ and every pair of complex $d\times d$ matrices $A_0,A_1$ whose rows and columns are indexed by $\{0,\ldots,d-1\}$, if $A_0^{*}A_0+A_1^{*}A_1=I_d$ and $A_0A_0^{*}+A_1A_1^{*}=I_d$, where $B^{*}$ denotes the conjugate transpose of a matrix $B$ and $I_d$ is the identity matrix, then there exist unitary complex $d\times d$ matrices $U,V$ and functions $x,y:\{0,\ldots,d-1\}\to\mathbb C$ such that simultaneously $VA_0U^{*}=\operatorname{diag}(x)$ and $VA_1U^{*}=\operatorname{diag}(y)$, where $\operatorname{diag}(x)$ has entry $x(i)$ at position $(i,i)$ and zero at every off-diagonal position, and likewise for $y$; unitary means that each matrix multiplied by its conjugate transpose in either order equals $I_d$. The dimension $d=0$ is excluded by the strict positivity hypothesis, while $d=1$ and matrices with zero entries or zero rank are included whenever the two displayed hypotheses hold, and no additional conditions are imposed on the complex diagonal entries $x(i),y(i)$.