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#1 2007-03-11 17:25:44

virtualinsanity
Member
Registered: 2007-03-11
Posts: 38

Orthogonality proofs

So I'm having trouble coming up with this proof, which I need to know for my quiz tomorrow, so any help would be greatly appreciated. 


An orthogonal matrix is defined as one for which A tranpose = inverse of A

In other words A^T = A^-1

Prove if a matrix A is orthogonal, then any two distinct columns of A have dot product zero.  Is the converse true?  Justify your answer.

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#2 2007-03-11 17:51:54

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Orthogonality proofs

Let the two columns be the ith and jth columns, where ij. Then the dot product of the columns is the product of the ith row of A[sup]T[/sup] with the jth column of A. This is the (i,j)th entry of the product A[sup]T[/sup]A = I, the identity matrix. Since  ij. this entry is 0.

The converse is certainly not true. Take A to be the zero matrix, for example.

Last edited by JaneFairfax (2007-03-11 17:56:09)

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#3 2007-03-12 09:20:32

lightning
Real Member
Registered: 2007-02-26
Posts: 2,060

Re: Orthogonality proofs

listion to jane shes really smart


Zappzter - New IM app! Unsure of which room to join? "ZNU" is made to help new users. c:

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#4 2007-03-12 18:11:23

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: Orthogonality proofs

kA is another example, where k is any nonzero scaler.


X'(y-Xβ)=0

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