Constraint on eigenvalues from matrix equations

Question

Given two 2 x 2 complex matrices $A$ and $B$ (not necessarily diagonalizable) such that they satisfy $$A^\dagger A+B^\dagger B=I.$$ is it possible to constrain the eigenvalues of matrices $A$ and $B$ using the above equality? Can someone give me some hints on how to do it? Thanks

Answer 1

Not too much can be said about the eigenvalues directly, but you can certainly get an upper bound. In particular: we note that for any vector $x$, we have $$ \|x\|^2 = x^\dagger(A^\dagger A + B^\dagger B)x = x^\dagger A^\dagger A x + x^\dagger B^\dagger Bx = \|Ax\|^2 + \|Bx\|^2 $$ So, we can conclude that for all vectors $x$, we have $\|Ax\| \leq \|x\|$ and $\|Bx\| \leq \|x\|$. So, any eigenvalue $\lambda$ of $A$ or $B$ must satisfy $|\lambda| \leq 1$.

Another notable property is that $\|Ax\| = \|x\| \iff \|Bx\| = 0$ (and symmetrically $\|Bx\| = \|x\| \iff \|Ax\| = 0$).

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About $A^\dagger B$: using the spectral norm, we have $$ \|A^\dagger B\| \leq \|A^\dagger\| \cdot \|B\| \leq 1 $$ which means that the absolute value of the eigenvalues of $A^\dagger B$ are at most $1$.