Find eigenvalues and eigenvectors of weird matrix

In summary: The first root is ##b = \pm \sqrt{2}## and the second root is ##b = -\lambda##.Ok so before i multiplied everything out, it came to:det|1 (a - lamnda)| + (a - lamnda)*det|(a - lamnda) 1 | = 0 |0 1 | |
  • #1
dukemiami
3
0

Homework Statement


find eigenvalues and eigenvectors for the following matrix

|a 1 0|
|1 a 1|
|0 1 a|

Homework Equations

The Attempt at a Solution


I'm trying to find eigenvalues, in doing so I've come to a dead end at 1 + (a^3 - lambda a^2 -2a^2 lambda + 2a lambda^2 + lambda^2 a - lambda^3 - a + lambda)

This is because i have to put in a - lamnda across the board, then it gets tricky when trying to find eigenvalues with all these variables, can someone please help in this bizarre question.
 
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  • #2
The polynomial you came up with isn't correct. I suggest you resist the temptation to multiply everything out immediately. You should find that ##(a-\lambda)## is a factor.
 
  • #3
Ok so before i multiplied everything out, it came to:

det|1 (a - lamnda)| + (a - lamnda)*det|(a - lamnda) 1 | = 0
|0 1 | | 1 (a - lamnda)|

which becomes 1 + (a - lamnda)((a - lamnda)(a - lamnda) - 1)

I think i see now, you're right with the a - lamnda

Thus a represents an eigenvalue, and is the only one then?
 
  • #4
dukemiami said:
Ok so before i multiplied everything out, it came to:

det|1 (a - lamnda)| + (a - lamnda)*det|(a - lamnda) 1 | = 0
|0 1 | | 1 (a - lamnda)|

which becomes 1 + (a - lamnda)((a - lamnda)(a - lamnda) - 1)

I think i see now, you're right with the a - lamnda

Thus a represents an eigenvalue, and is the only one then?

The

0 1 | | 1 (a - lamnda)|

should be shifted accordingly to match the top row
 
  • #5
I don't think your expansion is correct. To get matrices to appear correctly, use LaTeX. It's pretty easy to learn. If you reply to this post, you can see an example.
$$\begin{vmatrix}
a-\lambda & 1 & 0 \\
1 & a-\lambda & 1 \\
0 & 1 & a-\lambda
\end{vmatrix}$$
 
  • #6
dukemiami said:

Homework Statement


find eigenvalues and eigenvectors for the following matrix

|a 1 0|
|1 a 1|
|0 1 a|

Homework Equations

The Attempt at a Solution


I'm trying to find eigenvalues, in doing so I've come to a dead end at 1 + (a^3 - lambda a^2 -2a^2 lambda + 2a lambda^2 + lambda^2 a - lambda^3 - a + lambda)

This is because i have to put in a - lamnda across the board, then it gets tricky when trying to find eigenvalues with all these variables, can someone please help in this bizarre question.

If you set ##b = a - \lambda## the determinant of ##A - \lambda I## becomes ##b^3 -2b##, so equating this to zero gives roots ##b = 0## and ##b = \pm \sqrt{2}##.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to describe the behavior of a linear transformation. Eigenvalues represent the scaling factor of the eigenvectors, which are vectors that remain in the same direction after being transformed.

2. How do I find the eigenvalues and eigenvectors of a matrix?

To find the eigenvalues and eigenvectors of a matrix, you can use a variety of methods such as the characteristic polynomial, the power method, or the diagonalization method. The method used will depend on the size and complexity of the matrix.

3. Can I find eigenvalues and eigenvectors of any matrix?

Yes, you can find eigenvalues and eigenvectors of any square matrix. However, not all matrices will have real eigenvalues and eigenvectors. In some cases, complex eigenvalues and eigenvectors may be found.

4. What does it mean if a matrix has repeated eigenvalues?

If a matrix has repeated eigenvalues, it means that there are multiple eigenvectors associated with the same eigenvalue. This can occur when the matrix has a repeated factor in its characteristic polynomial or when the matrix has symmetry.

5. How are eigenvalues and eigenvectors used in real-world applications?

Eigenvalues and eigenvectors have various applications in fields such as physics, engineering, and data analysis. They can be used to study the stability of a system, understand the behavior of networks, and identify important features in data sets.

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