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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.

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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?

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  • $\begingroup$ oh sorry nevermind I figured it out. I was under the impression that if nonsquare matrices have SVDs, then the left-singular-matrix and right-singular-matrix had to be nonsquare. I just realized that in those cases, the diagonal matrix had to be nonsquare. My bad $\endgroup$ – woojoo666 Oct 30 '16 at 9:14
  • $\begingroup$ Is there a name for non square matrices which obey $ {X}^{T} X = I $? $\endgroup$ – Royi Jun 9 '18 at 20:48
  • $\begingroup$ @Royi semi-unitary $\endgroup$ – T. Fo Feb 5 at 19:09
  • $\begingroup$ Yep. In Wikipedia it is Semi Orthogonal Matrix. $\endgroup$ – Royi Feb 5 at 20:49

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