General Question(s) about Eigenvalue Problem

[Saladsamurai]

Homework Statement

I am having some trouble with this procedure and I am not exactly sure how to phrase my questions; so I will procede with one particular problem that is giving me trouble and perhaps someone can help to shed light on it.

In one problem, I am given some matrix

<br /> \begin{bmatrix}<br /> 1 &amp; 0 &amp; 1 \\<br /> 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; -2<br /> \end{bmatrix}<br />

and I am asked to find the eigenvalues (the lambdas) as well as the eigenvectors corresponding to them.

Homework Equations

Here I will just include a brief description of what "the eigenvalue problem" is. We are essentially looking for values of \lambda for which the equation (\mathbf{A} - \lambda\mathbf{I})\mathbf{x} = 0\qquad(1) admits nontrivial solutions (i.e. \mathbf{x}\ne0)

The Attempt at a Solution

Since the matrix A is upper triangular, it's eigenvalues are simply the elements along the main diagonal:

\lambda_{1,2,3} = \{1, 0, -2\}

For each of these, there is a corresponding eigenvector that we find by plugging each value back into (1).

For \lambda_1 = 1:

A =<br /> \begin{bmatrix}<br /> 0 &amp; 0 &amp; 1 \\<br /> 0 &amp; -1 &amp; 0 \\<br /> 0 &amp; 0 &amp; -3<br /> \end{bmatrix}<br /> \begin{bmatrix}<br /> x_1 \\<br /> x_2 \\<br /> x_3<br /> \end{bmatrix}<br /> =<br /> \begin{bmatrix}<br /> 0 \\<br /> 0 \\<br /> 0<br /> \end{bmatrix}<br /> \qquad(2)<br />

I am not really sure what to do with (2). At first glance, it suggests to me that all of the x_i's are zero. But the whole point of the eigenvalue problem is that all of the x_i's are not zero. So, how can I reconcile my system of equations with this?

The solution says that the eigenvector corresponding to \lambda_1 =1 is

<br /> e_1 = <br /> \begin{bmatrix}<br /> 1 \\<br /> 0 \\<br /> 0<br /> \end{bmatrix}<br />

How do they get this from (2)?

 

[Tedjn]

How do you find (a basis for) the nullspace (aka kernel) of a linear transformation?

 

[Saladsamurai]

Hi Tedjin I should have prefaced this with the fact that it is from a Mathematical Methods for Engineers class; so most of those words don't mean anything to me. I know what a basis is, but that's it. I'll definitely look them up now, but I think that the answer should be simpler or more intuitive then what you suggest. It appears that they let x2 = x3 = 0 and let x1 remain arbitrary, but I don't know why they chose x1 to be arbitrary over the others.

 

[fzero]

For \lambda_1 = 1:

A =<br /> \begin{bmatrix}<br /> 0 &amp; 0 &amp; 1 \\<br /> 0 &amp; -1 &amp; 0 \\<br /> 0 &amp; 0 &amp; -3<br /> \end{bmatrix}<br /> \begin{bmatrix}<br /> x_1 \\<br /> x_2 \\<br /> x_3<br /> \end{bmatrix}<br /> =<br /> \begin{bmatrix}<br /> 0 \\<br /> 0 \\<br /> 0<br /> \end{bmatrix}<br /> \qquad(2)<br />

I am not really sure what to do with (2). At first glance, it suggests to me that all of the x_i's are zero. But the whole point of the eigenvalue problem is that all of the x_i's are not zero. So, how can I reconcile my system of equations with this?

The solution says that the eigenvector corresponding to \lambda_1 =1 is

<br /> e_1 = <br /> \begin{bmatrix}<br /> 1 \\<br /> 0 \\<br /> 0<br /> \end{bmatrix}<br />

How do they get this from (2)?

(2) represents the set of equations

0 x_1 + 1 x_3 =0, ~ -1 x_2 = 0 ,~ 0 x1 - 3 x_3 =0.

You know how to solve these. There's no way to solve them with x_3 arbitrary.

 

[Saladsamurai]

(2) represents the set of equations

0 x_1 + 1 x_3 =0, ~ -1 x_2 = 0 ,~ 0 x1 - 3 x_3 =0.

You know how to solve these. There's no way to solve them with x_3 arbitrary.

Hi fzero I am not sure what you are driving at? Yes, (2) is the system:

x_3 = 0
-x_2 = 0
-3x_3 = 0

but I don't know how to solve this. It clearly suggests that x_2 = x_3 = 0, but there is no mention of x₁ in this system. Hmmm ... I think the light bulb just went off Since there is no mention of x₁ in this system, it means that it does not matter what value x₁ takes on ... it does not effect the system and hence it is arbitrary.


I know I have more question regarding eigenvalue problems, but I cannot think of what they are at the moment. So I will post back in a while with more. Thanks for your help so far.