Question

Eigenvectors

Answer 1

this is where i am up to but i think i am doing something wrong

Answer 2

Definitions


Let A be an nn matrix. The number is an eigenvalue of A if there exists a non-zero vector v such that
Av=v
In this case, vector v is called an eigenvector of A corresponding to .

Answer 3

you have left out something

Answer 4

\[\text{(c) Find and normalize the eigenvector (note that they are orthogonal).}\]
\[\text{------------------------------}\]

\[\textbf{T}|\alpha \rangle=\lambda_1|\alpha\rangle =-1|\alpha\rangle\]

\[ \begin{pmatrix} 1-(-1) & 1-i \\ 1+i & 0-(-1) \\ \end{pmatrix} \begin{pmatrix} \alpha_1\\ \alpha_2 \end{pmatrix} = -1\begin{pmatrix} \alpha_1\\ \alpha_2 \end{pmatrix}\]

\[ \begin{pmatrix} 2 & 1-i \\ 1+i & 1 \\ \end{pmatrix} \begin{pmatrix} \alpha_1\\ \alpha_2 \end{pmatrix} = \begin{pmatrix} -\alpha_1\\ -\alpha_2 \end{pmatrix}\]


\[2\alpha_1+(1-i)\alpha_2=-\alpha_1\quad{{(i)}}\]
\[(1+i)\alpha_1+\alpha_2=-\alpha_2\quad{{(ii)}}\]