Gram-Schimidt orthonormalization for three eigenvectors

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In summary, we have a matrix ##A## with three eigenvectors ##|\psi_1\rangle##, ##|\psi_2\rangle## and ##|\psi_3\rangle##, and we want to orthogonalize them to get ##|\phi_1\rangle##, ##|\phi_2\rangle##, and ##|\phi_3\rangle##. The process involves dividing each vector by its norm and subtracting the projections of the previous orthogonalized vectors. The third equation in this process can be checked for correctness by verifying that the inner product between ##\phi_3## and any of the previous vectors is equal to zero.
  • #1
Wrichik Basu
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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?
 
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  • #2
Wrichik Basu said:
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.
 
  • #3
Michael Price said:
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?
 
  • #4
Wrichik Basu said:
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:
 
  • #5
kuruman said:
Write them in such a way that they have different subscripts. :oldsmile:
:olduhh: Went tangentially past my brain. How would different subscripts matter?
 
  • #6
Wrichik Basu said:
:olduhh: Went tangentially past my brain. How would different subscripts matter?
You used the same subscript ##_1## for all three vectors.
 
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  • #7
Wrichik Basu said:
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}.##
 
  • #8
kuruman said:
If they have the same subscript, they are the same vector - it's just a typo fix.
Fixed typo.
 

Related to Gram-Schimidt orthonormalization for three eigenvectors

1. What is Gram-Schmidt orthonormalization?

Gram-Schmidt orthonormalization is a mathematical process used to transform a set of linearly independent vectors into a set of orthonormal vectors. This process involves finding the orthogonal projection of each vector onto the subspace spanned by the previous vectors and then normalizing the resulting vectors to have a length of 1.

2. Why is Gram-Schmidt orthonormalization important?

Gram-Schmidt orthonormalization is important because it allows us to transform a set of vectors into a set of orthonormal vectors, which are easier to work with in many mathematical applications. Additionally, orthonormal vectors have a number of useful properties that make them particularly useful in linear algebra and other areas of mathematics.

3. How does Gram-Schmidt orthonormalization work for three eigenvectors?

For three eigenvectors, the Gram-Schmidt orthonormalization process involves finding the orthogonal projection of the first eigenvector onto the subspace spanned by the other two eigenvectors. This projection is then subtracted from the first eigenvector to make it orthogonal to the other two. The same process is then repeated for the second and third eigenvectors, resulting in a set of three orthonormal eigenvectors.

4. What are the benefits of using Gram-Schmidt orthonormalization for three eigenvectors?

One of the main benefits of using Gram-Schmidt orthonormalization for three eigenvectors is that it allows us to transform a set of linearly independent eigenvectors into a set of orthonormal eigenvectors. This can simplify calculations and make it easier to work with these vectors in various mathematical applications. Additionally, orthonormal eigenvectors have a number of useful properties that can be leveraged in solving problems.

5. Are there any limitations to using Gram-Schmidt orthonormalization for three eigenvectors?

One limitation of Gram-Schmidt orthonormalization is that it can be computationally intensive, especially for larger sets of vectors. Additionally, this process may not always result in perfectly orthogonal or orthonormal vectors due to rounding errors or other issues. In some cases, alternative methods may be used to achieve similar results.

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