Gram-Schimidt orthonormalization for three eigenvectors

[Wrichik Basu]

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?

 

[Michael Price]

Say I have a matrix $A$ and it has three eigenvectors $|\psi_1\rangle$, $|\psi_1\rangle$ and $|\psi_1\rangle$. 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?

I suggest you edit your three original vectors.

 

[Wrichik Basu]

I suggest you edit your three original vectors.

You mean to say that I should write them such that they are orthogonal to each other?

 

[kuruman]

You mean to say that I should write them such that they are orthogonal to each other?

Write them in such a way that they have different subscripts.

 

[Wrichik Basu]

Write them in such a way that they have different subscripts.

Went tangentially past my brain. How would different subscripts matter?

 

[WWGD]

Went tangentially past my brain. How would different subscripts matter?

You used the same subscript $_1$ for all three vectors.

 

[kuruman]

Say I have a matrix $A$ and it has three eigenvectors $|\psi_1\rangle$, $|\psi_1\rangle$ and $|\psi_1\rangle$.

If they have the same subscript, they are the same vector - it's just a typo fix. You can check whether you have the correct expression for the third eigenvector by seeing if $\langle \phi_3|\phi_j\rangle\ = \delta_{3j}.$

 

[Wrichik Basu]

If they have the same subscript, they are the same vector - it's just a typo fix.

Fixed typo.