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An orthogonal matrix necessarily has orthonormal columns, and orthonormal columns necessarily give an orthogonal matrix. Also, orthonormal columns imply orthonormal rows.

But how about the converse of the last statement? Meaning, do orthonormal rows necessarily imply orthonormal columns?

Thanks

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    $\begingroup$ With square matrices, if $AB=I,$ then $BA=I$ as well. In your case, one matrix is some $C$ and the other $C^T.$ $\endgroup$ – Will Jagy Aug 13 '12 at 20:21
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do orthonormal rows necessarily imply orthonormal columns?

For square matrices, yes. To see this, note that if $M$ has orthonormal rows but not orthonormal columns then the transpose $M^T$ has orthonormal columns but not orthonormal rows, which you know to be impossible.

For non-square matrices we can have orthonormal rows but not orthonormal columns or vice-versa. For example, consider the matrix $$\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ \end{pmatrix}$$

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  • $\begingroup$ Yes, the context which I was considering was orthogonal (hence square) matrices. Edited the title to make that clear. Thanks! $\endgroup$ – Ryan Aug 13 '12 at 20:30

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