I understand that if A is orthogonal, then $A^{-1} = A^T$. However, i cannot seem to understand the proof.
Can someone put in some numbers in the proof. This site tries to explain it but i cannot seem to get $A^TA$ becoming an identity matrix.
Thanks.
Best Answer
Suppose $A$ an orthogonal matrix. Then we can write $A$ as $A = (a^1,a^2,...,a^n)$. With $a^i$ the column vectors of $A$.
Then we know that $a^1,a^2,...,a^n$ are pairwise orthogonal. E.g. $(a^i,a^j) = 0$ with $i≠j$ and $(a^i,a^j)= 1$ with $i=j$.
Now if you do the matrix multiplication $A^TA$ or $AA^T$ only the positions $a^{i,i}$ become ones, all others zeros.