Eigenvectors for degenerate eigenvalues

[dyn]

I am looking at some notes on Linear algebra written for maths students mainly to improve my Quantum Mechanics. I came across the following example - $$ \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix} $$
The example then gives the eigenvalues as 0 and 1(doubly degenerate). It then calculates the eigenvectors using Gaussian elimination. This is where my problem arises - coming from a physics background I tried to find the eigenvectors for the repeated eigenvalue 1 using back substitution but it doesn't seem to produce a solution this way. Am I doing something wrong or is it possible for back substitution not to work while Gaussian elimination works ?
The answer given for the eigenvector is a linear combination of the 2 vectors ( 3 1 0 )^T and (-1 0 1)^T. In the Quantum Mechanics textbook I am using it says for degenerate eigenvalues to choose 2 mutually orthogonal vectors. The 2 vectors I have listed are not orthogonal. Is the orthogonal part just a preference for QM and not a requirement ?
Thanks

 

[andrewkirk]

You can turn them into an orthogonal pair by subtracting from one the projection of the other onto it.

Given two linearly independent vectors $\vec u,\vec v$, the pair $\vec u-\frac{\vec u\cdot \vec v}{\vec v\cdot\vec v}\vec v, \vec v$ is orthogonal. You can check that by calculating $(\vec u-\frac{\vec u\cdot \vec v}{\vec v\cdot\vec v}\vec v)\cdot \vec v$

 

[dyn]

So choosing the eigenvectors as orthogonal is just a matter of preference. Thanks. Any thoughts on why I can't calculate the eigenvectors by back substitution but it can be done by Gaussian elimination ?

 

[dyn]

If I apply a general vector ( a b c )^T to the eigenvalue equation with eigenvalue 1 , I end up with 3 equations exactly the same a-3b+c=0. How do I then proceed to end up with the answer given which is equivalent to ( 3x-y , x , y )^T

 

[andrewkirk]

The equation 'a-3b+c=0' can be written as 'a=3b-c' which just says that for any an eigenvector with Eigenvalue 1, whose2nd and 3rd components are b,c, the first component is 3b-c.

Relabel a,b,c as x,y,z and you have the given answer.

 

[Mark44]

If I apply a general vector ( a b c )^T to the eigenvalue equation with eigenvalue 1 , I end up with 3 equations exactly the same a-3b+c=0. How do I then proceed to end up with the answer given which is equivalent to ( 3x-y , x , y )^T

Elaborating on what andrewkirk said, relabel the equation above as x - 3y + z = 0.

Then
x = 3y - z
y = y
z = ... z
If you look at the right sides as a sum of two vectors, you get
$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = y\begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} + z\begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$

Here y and z on the right side can be considered arbitrary constants.

 

[dyn]

Thanks for your replies. So essentially because I end up with 3 equations that are the same I really have just one equation with 3 unknowns. So I take 2 of those unknowns to have arbitrary values and the express the remaining unknown in terms of the 2 arbitrary values

 

[Mark44]

Thanks for your replies. So essentially because I end up with 3 equations that are the same I really have just one equation with 3 unknowns. So I take 2 of those unknowns to have arbitrary values and the express the remaining unknown in terms of the 2 arbitrary values

Yes. In the work I showed, you can take y = 1 and z = 0, and get one solution, and you can take y = 0, z = 1, to get another solution. Since y and z are completely arbitrary, you get a double infinity of solutions.

Geometrically, the two vectors I showed determine a plane in R³. Every point in this plane is some linear combination of those two vectors.