Answer to Question #328527 in Linear Algebra for wub

Question

25. For each of the linear operators below, determine whether it is normal, self adjoint, or neither (a) T : R2 → R2 defined by T(x, y) = (2x − 2y, −2x + 5y). (b) T : C 2 → C 2 defined by T(x, y) = (2x + iy, x + 2y). 

Answer 1

$a:\\T=\left[ \begin{matrix} 2& -2\\ -2& 5\\\end{matrix} \right] \\T^*=\left[ \begin{matrix} 2& -2\\ -2& 5\\\end{matrix} \right] =T\\TT^*=T^2=TT^*\\Normal, self-adjoint\\b:\\T=\left[ \begin{matrix} 2& i\\ 1& 2\\\end{matrix} \right] \\T^*=\left[ \begin{matrix} 2& 1\\ -i& 2\\\end{matrix} \right] \ne T\\TT^*=\left[ \begin{matrix} 2& i\\ 1& 2\\\end{matrix} \right] \left[ \begin{matrix} 2& 1\\ -i& 2\\\end{matrix} \right] =\left[ \begin{matrix} 5& 2+2i\\ 2-2i& 5\\\end{matrix} \right] \\TT^*=\left[ \begin{matrix} 2& 1\\ -i& 2\\\end{matrix} \right] \left[ \begin{matrix} 2& i\\ 1& 2\\\end{matrix} \right] =\left[ \begin{matrix} 5& 2+2i\\ 2-2i& 5\\\end{matrix} \right] =TT^*\\Normal, not\,\,self-adjoint$