Question

Question Regarding Matrices

Answer 1

Is it necessarily true that:
\[{\bf{Q}}^*{\bf{Q}} = {\bf{I}} \implies {\bf{QQ}}^* = {\bf{I}} \textrm{ if } {\bf{Q}} \in \mathbb{C}^{n x n}\]
Where:
\[{\bf{A}}^* \textrm{ denotes the conjugate transpose } {\bf{A}}^* = (\bar{{\bf{A}}})^T \textrm{ or better: } A_{jk}^*=\bar{A}_{kj}\]

Answer 2

Did you try taking the transpose of the above equations to see what happens? I haven't checked myself yet, but I believe the implication holds. Would have to type it out though.

Answer 3

If you take the transpose:
(Q*Q)* = Q*Q which doesn't help :(

Answer 4

(Q*Q)* = Q**Q*=QQ*=I since if Q have Q* that implied Q is self-adjoint, implied Q is invertible , in C^n, Q is unitary. right?