1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ Thanks a lot...so you are telling me that the concept orthonormality is applied only to vectors and not associated with matrices in general. $\endgroup$
    – Orpheus
    Sep 8 '20 at 0:25
  • $\begingroup$ 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. $\endgroup$
    – Erik D
    Sep 8 '20 at 0:28
  • $\begingroup$ 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 $QQ^T = I$. $\endgroup$
    – JimmyK4542
    Sep 8 '20 at 0:30
  • $\begingroup$ if a matrix $A$ is orthogonal then $A^T$ is the inverse of $A$ which implies $A$$A^T$= $A^T$$A$ = I $\endgroup$
    – Orpheus
    Sep 8 '20 at 0:41
  • $\begingroup$ @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. $\endgroup$
    – Erik D
    Sep 8 '20 at 0:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.