Calculating Eigenvectors and Eigenvalues for a Given Matrix

[Pr0grammer]

Homework Statement

Find the eigenvalues and the eigenvectors for the given matrix.

Homework Equations

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

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:
$$\[<br /> A+4I =<br /> \left[ {\begin{array}{ccc}<br /> 3 & 6 & 2 \\<br /> 0 & 9 & -6 \\<br /> 1 & 0 & 2 \\<br /> \end{array} } \right]<br /> \]$$ ~ $$\[<br /> \left[ {\begin{array}{ccc}<br /> 1 & 0 & 2 \\<br /> 0 & 1 & -2/3 \\<br /> 0 & 0 & 0 \\<br /> \end{array} } \right]<br /> \]$$

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

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

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

$$( A + 4 I ) \vec x =\[<br /> \left[ {\begin{array}{c}<br /> -2 \\<br /> 6 \\<br /> -2 \\<br /> \end{array} } \right]<br /> \]$$ - 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!


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

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 :)