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I don't understand what this example in my textbook is trying to show:

The matrix A = $\begin{bmatrix}1+i & 1 \\ 1-i & i\end{bmatrix}$ has orthogonal columns, but the rows are not orthogonal. Normalizing the columns gives the unitary matrix $\frac 12\begin{bmatrix}1+i & \sqrt{2} \\ 1-i & \sqrt{2}i\end{bmatrix}$

Is it saying that normalizing orthogonal columns of any matrix will produce a unitary matrix with orthonormal set of columns and orthonormal set of rows? Also, does this imply that unitary matrices must have orthonormal set of columns and rows?

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  • $\begingroup$ If you only consider square matrices, then yes. $\endgroup$
    – user251257
    Apr 1, 2016 at 15:22
  • $\begingroup$ @user251257 Ok thanks, then does a square complex matrix with orthonormal set of columns and rows imply that the matrix is unitary? $\endgroup$
    – sucksatmath
    Apr 1, 2016 at 15:34
  • $\begingroup$ @sucksatmath yes. $\endgroup$
    – Ben Grossmann
    Apr 1, 2016 at 19:19
  • $\begingroup$ @Omnomnomnom What if the matrix only has orthonormal set of columns (it's unknown if it has orthonomal rows)? $\endgroup$
    – sucksatmath
    Apr 1, 2016 at 20:03
  • $\begingroup$ If $A$ has an orthonormal set of columns, then $A^*A = I$. That is, $A^* = A^{-1}$. It follows that $AA^* = I$, which is to say that $A$ has an orthonormal set of rows. $\endgroup$
    – Ben Grossmann
    Apr 1, 2016 at 20:04

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