Using the definition of unitary / orthogonal operators explicity for matrices:

Question

If A is unitary, then $$AA^* = A^*A = I, and\ A^* = A^{-1}$$

I want to see this explicitly for a very simple unitary matrix, say, take the column vector A = (1,0,0) and we regard this as a 3x1 orthogonal matrix.

But $AA^*$ gives a 3x3 matrix which is not the identity matrix but rather a matrix with a 1 in the entry $a_{11}$ and zeros everywhere else in the matrix.

So, where did things go "wrong"?

Thanks,

Answer 1

You need that $A$ is a square matrix, not a column.

Edit: Omnomnomnom was first :)