Orthogonal Sets

  • #1

Homework Statement



Let U be an mxn matrix with orthonormal columns, and let x and y be in R^n. Prove

(Ux).(Uy)=x.y

Homework Equations



I know ||Ux||^2=(Ux)^T(Ux)

The Attempt at a Solution



I just have no idea...
 

Answers and Replies

  • #2
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Do you know anything about inverses of orthonormal matrices?
 
  • #3
I know that if U is orthogonal then (U^T)(U)=I, and by the Invertible Matrix Theorgem U inverse = U transpose?
 
  • #4
128
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Yes, use this together with Ux = x^T U^T, matrix multiplication is associative and that x . u = x^T u.
 
  • #5
Okay, I get this far:
(Ux).(Uy)=(x^T)(U^T).(y^T)(U^T)

It's like drawing blood from a stone - I get stuck at every step...
 
  • #6
128
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You did one steep too far.

(Ux)(Uy) = (x^T)(U^T)(Uy) = (x^T)[(U^T)(U)](y).

Now use what you know about (U^T)(U).
 
  • #7
Dick
Science Advisor
Homework Helper
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Do you know anything about inverses of orthonormal matrices?

I don't think you can really do it that neatly. U is mxn, not nxn. It's not necessarily a square matrix. It doesn't necessarily have an inverse. Try working on the usual basis of R^n={e1,...,en}. U(e_i) is the ith column of U. You have to be a little more detail oriented here.
 
  • #8
Oh, then we have (x^T)(y)=x.y

I didn't realise you could break up the dot product components and just multiply them together
 
  • #9
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I don't think you can really do it that neatly. U is mxn, not nxn. It's not necessarily a square matrix. It doesn't necessarily have an inverse. Try working on the usual basis of R^n={e1,...,en}. U(e_i) is the ith column of U. You have to be a little more detail oriented here.

Ah shiet. That is right.. I should have read more carefully.

To OP, What I have said so far works for orthonormal matrices but that is not what you have lol. Sorry.
 
  • #10
Dick
Science Advisor
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Ah shiet. That is right.. I should have read more carefully.

To OP, What I have said so far works for orthonormal matrices but that is not what you have lol. Sorry.

Carry on. It's easy to fix. U(e_i).U(e_j) is 1 if i=j and zero otherwise. Because they just two different columns of U. U^T*U is still the nxn identity matrix. You just can't call it the inverse. That's all I meant to clarify. You were doing fine otherwise.
 
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