B Gram-Schimidt orthonormalization for three eigenvectors

Wrichik Basu

Insights Author
Gold Member
2018 Award
1,106
961
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 learnt from this video, where the professor has done upto Eqn. (2) only. Can you check whether the third equation above is correct or not?
 
Last edited:
301
81
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 learnt 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

Insights Author
Gold Member
2018 Award
1,106
961
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

Science Advisor
Homework Helper
Insights Author
Gold Member
8,132
1,769
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. :oldsmile:
 

Wrichik Basu

Insights Author
Gold Member
2018 Award
1,106
961
Write them in such a way that they have different subscripts. :oldsmile:
:olduhh: Went tangentially past my brain. How would different subscripts matter?
 

WWGD

Science Advisor
Gold Member
4,300
1,853

kuruman

Science Advisor
Homework Helper
Insights Author
Gold Member
8,132
1,769
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}.##
 

Want to reply to this thread?

"Gram-Schimidt orthonormalization for three eigenvectors" You must log in or register to reply here.

Related Threads for: Gram-Schimidt orthonormalization for three eigenvectors

Replies
1
Views
2K
  • Posted
Replies
2
Views
6K
  • Posted
Replies
5
Views
1K
Replies
2
Views
3K
Replies
4
Views
2K
Replies
1
Views
9K
Replies
2
Views
235
  • Posted
Replies
1
Views
6K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top