Question

what condition must a and b satisfy for matrix [a+b b-a a-b b+a] to be orthogonal? I'm desperate to find the solution, i don't understand all of this

Answer 1

$$Q=\left[\begin{matrix}
a+b&b-a\\
a-b&b+a\end{matrix}\right]$$
$$Q^T=\left[\begin{matrix}
a+b&a-b\\
b-a&b+a\end{matrix}\right]$$

to become orthogonal,
$$QQ^T=Q^TQ=I$$( http://en.wikipedia.org/wiki/Orthogonal_matrix)

$$\begin{align*}QQ^T&=\left[\begin{matrix}
a+b&b-a\\
a-b&b+a\end{matrix}\right]\left[\begin{matrix}
a+b&a-b\\
b-a&b+a\end{matrix}\right]\\&=\left[\begin{matrix}
(a+b)^2+(b-a)^2 & (a^2-b^2)+(b^2-a^2)\\
(a^2-b^2)+(b^2-a^2)&(a-b)^2+(b+a)^2\end{matrix}\right]\\
&=\left((a+b)^2+(b-a)^2 \right)\left[\begin{matrix}
1&0\\
0&1\end{matrix}\right]\end{align*}$$
for this to become I,$$(a+b)^2+(b-a)^2=1\implies 2a^2+2b^2=1$$

Answer 2

thank you! i understand