Calculating Eigenvectors and Eigenvalues for a Given Matrix

[Pr0grammer]

Homework Statement

Find the eigenvalues and the eigenvectors for the given matrix.

Homework Equations

$$\[ A = \left[ {\begin{array}{ccc} -1 & 6 & 2 \\ 0 & 5 & -6 \\ 1 & 0 & -2 \\ \end{array} } \right] \]$$

The Attempt at a Solution

I solved $$A-\lambda I = 0$$ and got eigenvalues of -4 and 3, which I've confirmed as correct. After that, to solve for the eigenvalue of -4:
$$\[ A+4I = \left[ {\begin{array}{ccc} 3 & 6 & 2 \\ 0 & 9 & -6 \\ 1 & 0 & 2 \\ \end{array} } \right] \]$$ ~ $$\[ \left[ {\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -2/3 \\ 0 & 0 & 0 \\ \end{array} } \right] \]$$

so $$\vec x = a \[ \left[ {\begin{array}{c} -2 \\ 2/3 \\ 1 \\ \end{array} } \right] \]$$ for all a≠0.

...however, according to two different calculators, $$\vec x = a \[ \left[ {\begin{array}{c} 1.1872 \\ -.3957 \\ -.5936 \\ \end{array} } \right] \]$$

...which I've verified as being a working solution, while with mine:

$$( A + 4 I ) \vec x =\[ \left[ {\begin{array}{c} -2 \\ 6 \\ -2 \\ \end{array} } \right] \]$$ - all values should equal zero, which they do with the calculated solution. What am I doing wrong?

 

[drag12]

I think you've just made a calculation error while checking your work. I just did the multiplication and (A+4I)x=0. Somehow when you multiplied the matrix by x you came up with just the negative of the third column in (A+4I). Try it again and if you're still having the same problem try googling "Matrix Multiplication" and make sure you're using the correct process.

 

[Gregg]

Your eigenvector works for the eigenvalue -4. You need to do the same for eigen value 3, I don't think it's possible to find 3 L.I. eigenvectors though!


$$\left(\begin{array}{c} 1 \\ \frac{3}{2} \\ 0 \end{array} \right)$$

that is a "nice" vector to put in P though!

 

[Pr0grammer]

Doh. I entered 0 in my calculator instead of 1 for the last value of x. Thanks anyways :)