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I know that the columns of an orthogonal matrix are perpendicular to each other and additionally if the columns have unit length then they are orthonormal. But my professor states that the columns of an orthogonal matrix form an orthonormal basis? Is this right? Then what is the difference between orthogonal and orthonormal matrix?

Orpheus
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1 Answers1

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An orthogonal matrix may be defined as a square matrix the columns of which forms an orthonormal basis. There is no thing as an "orthonormal" matrix.

The terminology is a little confusing, but it is well established.

Erik D
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  • Thanks a lot...so you are telling me that the concept orthonormality is applied only to vectors and not associated with matrices in general. – Orpheus Sep 08 '20 at 00:25
  • Yes. The concepts of orthogonality and orthonormality are defined for vectors. Then the word "orthogonal" is used again to denote a (related, but slightly different) property of matrices - which often is a source of confusion for beginners. – Erik D Sep 08 '20 at 00:28
  • I've seen the term "orthonormal matrix" used to me a matrix whose columns are orthonormal vectors. So an orthonormal matrix $Q$ satisfies $Q^TQ = I$ but not necessarily $QQ^T = I$. – JimmyK4542 Sep 08 '20 at 00:30
  • if a matrix $A$ is orthogonal then $A^T$ is the inverse of $A$ which implies $A$$A^T$= $A^T$$A$ = I – Orpheus Sep 08 '20 at 00:41
  • @Orpheus: Yes. So with Jimmy's definition, "orthonormal matrix" is a weaker concept than "orthogonal matrix" (every orthogonal matrix is orthonormal, but not the other way around).

    I have never seen the word orthonormal used in that way, though.

     – Erik D Sep 08 '20 at 00:44
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    @Erik Don't the rows also form an orthonormal basis? – heretoinfinity Jul 05 '23 at 16:37