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Somefantastik
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Homework Statement
Choose [tex]\lambda_{1}, \lambda_{2}, \lambda_{3} [/tex] along with a set of vectors [tex]{v_{1},v_{2},v_{3}}[/tex] and construct an Hermitian matrix H with the eigenpairs [tex] (\lambda_{1},v_{1}),(\lambda_{2},v_{2}),(\lambda_{3},v_{3})[/tex]
Homework Equations
The Attempt at a Solution
[tex] \lambda_{1} = 2[/tex]
[tex]\lambda_{2} = -2[/tex]
[tex]v_{1} = \widehat{i}+2\widehat{j}[/tex]
[tex]v_{2} = \widehat{i}-2\widehat{j}[/tex]
[tex]u_{1} = \frac{1}{\sqrt{5}}\left(\widehat{i} + 2\widehat{j} \right)[/tex]
[tex]u_{2} = \frac{1}{\sqrt{5}}\left(\widehat{i} - 2\widehat{j}\right)[/tex]
[tex]v_{3} = u_{1} \times u_{2} = -\frac{4}{5}\widehat{k}[/tex]
[tex]u_{3} = -\widehat{k}[/tex]
[tex]Basis: \ U = \left\{ u_{1},u_{2},u_{3} \right}\}[/tex]
[tex]H = \left| \begin{array}{ccc} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1 \end{array} \right| [/tex]
How does this look?