# Find eigenvalue for matrix B= {[3,4,12],[4,-12,3],[12,3,-4]}

1. May 12, 2013

### LosTacos

1. The problem statement, all variables and given/known data

Find eigenvalue for matrix B= {[3,4,12],[4,-12,3],[12,3,-4]}

2. Relevant equations

3. The attempt at a solution

I set up the charactersitic polynomial and got the equation:
Pa(x) = (x-3)(x+12)(x+4) = x3 + 132 - 144 + 144 = x3 + 132

So I have 3 eigenvalues: 0, 13, -13. Is this correct?

2. May 12, 2013

### LosTacos

Actually I forgot to calculate teh determinants so I got:

Pa(x) = (x-3)(x+12)(x+4) + 2197 = x3 + 13x2 + 2053

3. May 12, 2013

### hsetennis

Those are the correct eigenvalues (if it is conventional to include 0 in your class). It may be good to note multiplicity of eigenvalues.

4. May 12, 2013

### LosTacos

From the characteristic polynomial x3 + 13x2 + 2053, how do I get the correct eigenvalues?

5. May 12, 2013

### Staff: Mentor

Two of them are correct, but 0 is not an eigenvalue. It should be -13 (a repeated eigenvalue).

You can't, since this isn't the correct characteristic polynomial. The eigenvalues are the roots of the characteristic polynomial.

6. May 12, 2013

### LosTacos

Okay I understand. So my eigenvalues are 13 and -13. If I was asked to find the basis for both of these, how do I go about doing that. I tried to solve the equation [13I2 - A I 0] however if ran into a wall. I row reduced it to get the matrix = {[2, 2/5, 6/5], [0,1,3], [0,0,1]}. But I wasnt sure where to go from there

7. May 12, 2013

### Staff: Mentor

You should be solving the matrix equation (13I3 - A)x = 0, or equivalently, (A - 13I)x = 0.
This would be the work to find the eigenvector for λ = 13.
This is incorrect. Show me the matrix you started with, and a step or two of your work.

8. May 12, 2013

### LosTacos

(A - 13I)x = 0:

{[10, 4, 12, 0],[4, 1, 3, 0],[12, 3, 17]} = {[1, 2/5, 6/5],[4, 1, 3, 0],[12, 3, 17, 0]} = {[1, 2/5, 6/5, 0], [0, -3/5, -9/5, 0],[0, -9/5, 13/5, 0]} = {[1, 2/5, 6/5, 0],[0, 1, 3, 0], [0, 0, 8, 0]}

9. May 13, 2013

### Staff: Mentor

You started off with an incorrect matrix.

Here is A:
$$\begin{bmatrix} 3 & 4 & 12\\ 4 & -12 & 3 \\ 12 & 3 &-4 \end{bmatrix}$$
To get A - 13I, subtract 13 from the entries on the main diagonal.