# Proving a matrix is orthogonal.

1. Apr 29, 2012

### Lucy Yeats

1. The problem statement, all variables and given/known data

Question 10a of the attached paper.

2. Relevant equations

3. The attempt at a solution

If a matrix is orthogonal, its transpose is its inverse.

The inverse $U^{-1}$ is defined by Ʃ$U^{-1}$ij Vj = uj

I don't know how to go about proving this. Thanks for any help!

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2. Apr 29, 2012

### HallsofIvy

Staff Emeritus
There is no attached paper! Also your given definition of "inverse" can't be true because it makes no sense- you haven't said what Vj and uj are (and you must mean ui, not uj because you should be summing over j). If you are given a specific matrix, you need to answer three questions:
1) What is its transpose?
2) What is its inverse?
3) Are they the same?

3. Apr 29, 2012

### Lucy Yeats

I attached the paper about a minute after I posted; I think it's there now. :-)

4. Apr 29, 2012

### Lucy Yeats

Also, the sum is from j=1 to n. So the ith element of the vector u is the sum of the elements of one row of the matrix U with the elements of the vector j.

5. Apr 29, 2012

### Office_Shredder

Staff Emeritus
Take the inner product of vi and vj using their expansion in terms of u's, and consider how your answer relates to matrix multiplication