A is an orthogonal matrix, 0<a<1 is a scalar and (aA+(1-a)I) is also orthogonal. I need to find A.
I'm trying to develop the expression (aA+(1-a)I)(aA+(1-a)I)^t = I, but that doesn't get me anywhere.
Hmm...this is what I get:
$(aA + (1-a)I)^T(aA + (1-a)I) = I$
$(aA^T + (1-a)I)(aA + (1-a)I) = I$
$a^2A^TA + (a - a^2)(A^T + A) + (1-a)^2I = I$
$(a - a^2)(A^T + A) = I - a^2I - I + 2aI - a^2I = 2(a - a^2)I$.
Since $0 < a < 1$, we have $a - a^2 \neq 0$, so that:
$A^T + A = 2I$.
What does this say about what the diagonal elements of $A$ must be?
Finally, if $A_j$ is the $j$-th column of $A$, show that $\|A_j\| = 1$ (since $A$ is orthogonal). What does this force the off-diagonal elements to be?