Clear difference between unitary matrix and semi-unitary matrices?

Question

What are the clear differences between Unitary and Semi-unitary.

Answer 1

A unitary matrix $U_{n\times n}$ is generally a complex matrix $U$ whose columns (or rows) constitute an orthonormal basis for $\mathbb C^n$. This means $$U^*U=UU^*=I$$ It is also invertible with $U^{−1} = U^∗$.

A semi-unitary matrix $U_{m\times n}$ is a non-square matrix ($m>n$ or $m<n$) where $U^*U=I_n$ or $UU^*=I_m$ which means either the rows or columns of the matrix are orthonormal.

Answer 2

Unitary Matrix: where $U^\dagger U=I$ (the Hermitian of $U$ times $U$ equaling the identity matrix)

Semi-Unitary Matrix: where $U^TU=I$ (the transpose of $U$ times $U$ equaling the identity matrix)

I'll be updating this with resources as I find them. Hope this helps!