Proving a matrix is orthogonal.

[Lucy Yeats]

Homework Statement

Question 10a of the attached paper.

Homework Equations

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!

 

[HallsofIvy]

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?

 

[Lucy Yeats]

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

 

[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.

 

[Office_Shredder]

Take the inner product of v_i and v_j using their expansion in terms of u's, and consider how your answer relates to matrix multiplication