[Math] Orthogonal matrix confusion

linear algebra

I have a confusion about orthogonal matrix.

  1. If columns of a square matrix are orthonormal to each other, is the matrix orthogonal?
  2. If yes, then are the rows of the matrix also orthonormal? Why?
  3. Why is it that QQ'=I? I get Q'Q=I but why QQ' is also I?

Thanks,
Tom

Best Answer

The key to your last question is that if a matrix $A$ has inverse $B$, then $$ AB = BA = I $$ Where $I$ is the identity matrix. That is, a finite square matrix always commutes with its inverse. From there, it's clear that if $Q'Q=I$ (that is, $Q'$ is the inverse of $Q$), then $QQ'=I$

The easy answer to your second question is that $Q$ is an orthogonal matrix whenever $Q'Q=I$, which means that $QQ'=I$. Now, if $Q$ is an orthogonal matrix, then $$(Q')'Q'=QQ'=I$$ which means that $Q'$ is orthogonal. This in turn means that $Q'$ has orthonormal columns, which means that $Q$ has orthonormal rows.

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