Show that A is an orthogonal matrix

[yoghurt54]

Homework Statement

If {a_j} and {b_j} are two separate sets of orthonormal basis sets, and are related by

a_i = \sum_jⁿA_ijb_j

Show that A is an orthogonal matrix

Homework Equations

Provided above.

The Attempt at a Solution

Too much latex needed to show what I tried, but basically I considered the properties of an orthogonal matrix: AA^T = 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 a_j.a_k = o if j and k are different and 1 if they are the same, and the same for the vectors b_j.

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

 

[HallsofIvy]

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.