Are there infinitely many unitary matrices?

Question

Will there be infinitely many unitary matrices of the form of $U_{n \times n}$ with complex coefficients?

How about unitary matrices of the same form but with only real coefficients?

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(Not a homework question, just a curious thought).

Answer 1

The unitary matrices with real coefficients are the orthogonal matrices. That is, $O(n)\subset U (n)$.

In dimension two, there are uncountably many orthogonal matrices, the rotations: $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$, for $\theta\in [0,2\pi)$, forming $SO(2)\subset O(2)$.

$SO(2)$ is thus isomorphic to the circle group, sometimes denoted $U(1)$.

You can use induction to prove it for $n\gt2$, as suggested in the comments by @lhf.

Answer 2

Any rotation matrix is unitary, and there is an uncountable infinity even of $2 \times 2$ kind.