Finding Orthogonal Set

Example with Matrixes

• With $A=\left[\begin{array}{cc}2& 1\\ -2& 0\end{array}\right]$
• Then, $⟨A|B⟩=trace(A{B}^{\ast })$
• Let $W=span\{A\}$
• Then, find $W^{\perp }$
• We consider $W^{\perp }=\{B|⟨B|aA⟩=0\}$
• $⟹\{B|trace(Ba{A}^{\ast })=0\}$
• With $B=\left[\begin{array}{cc}{b}_1& b_2\\ b_3& b_4\end{array}\right]$

2ab_{1} + ab_{2} & -2ab_{1}\ 2ab_{3} + ab_{4} & -2ab_{3}\ \end{array}\right]$$

• Then, we find $tr(Ba{A}^{\ast })=0⟹2a{b}_1+a{b}_2-2a{b}_3=0$
• Thus, $b_1=b_3-\frac{1}{2}{b}_2$
• Then, we can form a matrix: $B=\left[\begin{array}{cc}{b}_3-\frac{1}{2}{b}_3& b_2\\ & \\ b_3& b_4\end{array}\right]$
• \left[\begin{array}{cc}

b_{3} & 0 \ b_{3} & 0 \ \end{array}\right] + \left[\begin{array}{cc} \frac{1}{2 } b_{2} & b_{2} \ 0 & 0 \ \end{array}\right] + \left[\begin{array}{cc} 0 & 0 \ 0 & b_{4} \ \end{array}\right] +

- Finally, $$ W^{\perp} = span \{ \left[\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ \end{array}\right] , \left[\begin{array}{cc} \frac{1}{2 } & 1 \\ 0 & 0 \\ \end{array}\right] , \left[\begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array}\right] \}