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).
a:T=[2−2−25]T∗=[2−2−25]=TTT∗=T2=TT∗Normal,self−adjointb:T=[2i12]T∗=[21−i2]≠TTT∗=[2i12][21−i2]=[52+2i2−2i5]TT∗=[21−i2][2i12]=[52+2i2−2i5]=TT∗Normal,not self−adjointa:\\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-adjointa:T=[2−2−25]T∗=[2−2−25]=TTT∗=T2=TT∗Normal,self−adjointb:T=[21i2]T∗=[2−i12]=TTT∗=[21i2][2−i12]=[52−2i2+2i5]TT∗=[2−i12][21i2]=[52−2i2+2i5]=TT∗Normal,notself−adjoint
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