# Non Square Unitary Matrices

In numerous sources online, I have found three claims

• Unitary matrices follow $U^* = U^{-1}$, where $^*$ is the conjugate transpose
• Unitary matrices can be non-square, as long as all columns and rows are orthonormal
• Non-square matrices do not have inverses

These statements seem to form a contradiction, though I feel like it has something to do with complex numbers. Can anybody help clarify the statements above, and provide an example of a non-square unitary matrix?

Correction

After copious amounts of google searching, I forgot that the second statement was an assumption I made. I was under the impression that if non-square matrices have SVDs, then the left-singular-matrix and right-singular-matrix had to be non-square. I just realized that in those cases, they are actually square, and it is the diagonal matrix that is non-square. My bad.

The first statement is certainly true since it is a possible definition of unitary matrices. In my opinion, it is not possible to define non-square unitary matrices because how would you define a unity matrix in the space $L(m,n)$, i.e. the space of linear maps from a $m$-dimensional to a $n$-dimensional vector space? Where did you get that claim from?
• Is there a name for non square matrices which obey ${X}^{T} X = I$? – Royi Jun 9 '18 at 20:48