Show that A is an orthogonal matrix

  • Thread starter yoghurt54
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  • #1
yoghurt54
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Homework Statement





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

ai = [tex]\sum[/tex]jnAijbj

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: 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.
 

Answers and Replies

  • #2
HallsofIvy
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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.
 

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