Math Is Fun Forum

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.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Orthogonality proofs

Let the two columns be the ith and jth columns, where i ≠ j. 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  i ≠ j. 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)

lightning

Real Member

Registered: 2007-02-26

Posts: 2,060

Re: Orthogonality proofs

listion to jane shes really smart

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George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Orthogonality proofs

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

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X'(y-Xβ)=0