Find the matrix $B\in \mathbb{C}^{n \times n}$

Question

Suppose $A\in \mathbb{C}^{n \times n}$ be a given unitary matrix then if there exists a non zero matrix $B$ so that $A+B$ is an unitary matrix then $B$ needs to satisfy the equation $A^*B+B^*A+B^*B=0$. How to find the matrix $B$ which satisfy this equation?

Answer 1

Lord Shark the Unknown had it in his comment; if $U \ne A$ is any other unitary,

$U^\dagger U = U U^\dagger = I; \tag 1$

then setting

$B = U - A \ne 0, \tag 2$

we have

$A^\dagger B + B^\dagger A + B^\dagger B = A^\dagger (U - A) + (U - A)^\dagger A + (U - A)^\dagger (U - A)$ $= A^\dagger U - A^\dagger A + U^\dagger A - A^\dagger A + (U^\dagger - A^\dagger)(U - A)$ $= A^\dagger U - A^\dagger A + U^\dagger A - A^\dagger A + U^\dagger U - U^\dagger A - A^\dagger U + A^\dagger A$ $= A^\dagger U - I + U^\dagger A - I + I -U^\dagger A - A^\dagger U + I = 0; \tag 3$

so there are indeed very many $B \ne 0$ satisfying

$A^\dagger B + B^\dagger A + B^\dagger B = 0, \tag 4$

one $B \ne 0$ for each unitary $U \ne A$.

In fact, all this stuff works with $U = A$, $B = 0$ as well.

Note Added in Edit, Monday 10 September 2018 12:19 PM PST: This in response to our OP Saheb's inquiry, expressed below his comment to this answer, as to whether and what might be necessary and sufficient conditions for the existence of $0 \ne B \in \Bbb C^{n \times n}$ with $A + B$ unitary. For $n \ge 1$, a sufficient condition is simply the mere existence of unitary $A \in \Bbb C^{n \times n}$; for we may choose $\theta \in (0, 2\pi)$ yielding $e^{i\theta} \ne 1$; then $U = e^{i\theta} A \ne A$ is unitary as well

$(e^{i\theta}A)^\dagger (e^{i\theta}A) = e^{-i\theta}e^{i\theta}A^\dagger A = A^\dagger A = I; \tag 5$

so setting $B = U - A$ we have $A + B$ unitary; $B$ will then satisfy equation (4) as we have seen; and of course, if $B$ satisfies (4) and $A + B$ is unitary, then

$A^\dagger A = A^\dagger A + A^\dagger B + B^\dagger A + B^\dagger B = (A + B)^\dagger (A + B) = I, \tag 6$

so $A$ must be unitary as well. End of Note.