Question

Consider linear operator T:C^3-->C^3 def by T(z1,z2,z3)=(z1+iz2, iz1-2z2, -iz2+z3). Check T* and check if T is self adjoined. check if T is unitary

Answer 1

Here matrix representation of T with respect to standard basis $\{ (1,0,0), (0,1,0), (0,0,1)\}$ over the field $C$ is $A=$

$\begin{pmatrix} 1& i& 0 \\ i & -2&0\\ 0&-i&1 \end{pmatrix}$

Now , $T^*=$ Transpose conjugate of matrix representation of $T$

$=A^H$

$=$ $\begin{pmatrix} 1&-i&0 \\ -i&-2&0\\ 0&i&1 \end{pmatrix}$

Since ,$A\neq A^H$

Hence, $T$ is not a self adjoint operator.

To check unitary operators ,

Consider $A.A^H=$ $\begin{pmatrix} 2&-3i&-1\\ 3i&5&-2i\\ -1&2i&2 \end{pmatrix}$ $\neq I$

Where $I$ stands for identity matrix.

Therefore,$T$ is not a unitary operator.