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Say I have a matrix ##A## and it has three eigenvectors ##|\psi_1\rangle##, ##|\psi_2\rangle## and ##|\psi_3\rangle##. I want to orthogonalize these. Say my orthogonalized eigenvectors are ##|\phi_1\rangle##, ##|\phi_2\rangle## and ##|\phi_3\rangle##.

$$

\begin{eqnarray}

|\phi_1\rangle = \frac{|\psi_1\rangle}{|| \psi_1 ||} \\

\text{ } \nonumber \\

\phi_2 = \dfrac{ \left\{| \psi_2\rangle - \langle\phi_1 | \psi_2\rangle |\phi_1\rangle \right\} }{|| \{ \cdots \} ||} \\

\text{ } \nonumber \\

\phi_3 = \dfrac{ \left\{| \psi_3\rangle - \langle\phi_1 | \psi_3\rangle |\phi_1\rangle - \langle\phi_2 | \psi_3\rangle |\phi_2\rangle \right\} }{|| \{ \cdots \} ||}

\end{eqnarray}$$

Our teacher didn't explain this at all. I learned from this video, where the professor has done upto Eqn. (2) only. Can you check whether the third equation above is correct or not?

$$

\begin{eqnarray}

|\phi_1\rangle = \frac{|\psi_1\rangle}{|| \psi_1 ||} \\

\text{ } \nonumber \\

\phi_2 = \dfrac{ \left\{| \psi_2\rangle - \langle\phi_1 | \psi_2\rangle |\phi_1\rangle \right\} }{|| \{ \cdots \} ||} \\

\text{ } \nonumber \\

\phi_3 = \dfrac{ \left\{| \psi_3\rangle - \langle\phi_1 | \psi_3\rangle |\phi_1\rangle - \langle\phi_2 | \psi_3\rangle |\phi_2\rangle \right\} }{|| \{ \cdots \} ||}

\end{eqnarray}$$

Our teacher didn't explain this at all. I learned from this video, where the professor has done upto Eqn. (2) only. Can you check whether the third equation above is correct or not?

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