# Show that A is an orthogonal matrix

1. Jul 22, 2010

### yoghurt54

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

If {aj} and {bj} are two separate sets of orthonormal basis sets, and are related by

ai = $$\sum$$jnAijbj

Show that A is an orthogonal matrix

2. Relevant equations

Provided above.

3. The attempt at a solution

Too much latex needed to show what I tried, but basically I considered the properties of an orthogonal matrix: AAT = I and considered which elements would multiply and sum to give the 1's and 0's of the identity matrix.

I then considered the fact that aj.ak = o if j and k are different and 1 if they are the same, and the same for the vectors bj.

Then I thought of the property of preserving norms, but can't see how to connect it to this problem.

2. Jul 22, 2010

### HallsofIvy

Staff Emeritus
More to the point is that an orthonormal matrix is a matrix whose columns (or rows), thought of as vectors, are orthonormal- that is, each has length 1 and any two are orthogonal.