- #1
jeff.berhow
- 17
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Homework Statement
Determine a basis for each eigenspace and whether or not the matrix is defective.
\begin{array}{ccc}
3 & -4 & -1 \\
0 & -1 & -1 \\
0 & -4 & 2 \end{array}
Homework Equations
Regular ol' eigenvector, eigenvalue business.
The Attempt at a Solution
Ok, so I've found the eigenvalues first with no problem. λ_1 = 3 (with multiplicity 2) and λ_2 = -2. My misunderstanding comes with finding the eigenvectors with λ_1. Gauss-Jordan with the new augmented matrix gives me:
\begin{array}{ccc}
0 & 1 & 1/4 & | & 0\\
0 & 0 & 0 & | & 0\\
0 & 0 & 0 & | & 0\end{array}
Let x_3 = r and this tells me that x_1 = 0r, x_2 = -1/4r and so the eigenvector is (0, -1/4, 1). This then leads me to believe the matrix will be defective as I will only be able to get 1 more eigenvector out of λ_2 = -2 as this matrix has been exhausted.
Lo and behold! In the back of the book, they were able to extract another vector out of the matrix above: (1, 0, 0). This would then give us three total eigenvectors which is indeed a basis and makes the matrix nondefective. So, my question is, how did they get another vector out of that thing?
Thanks in advance,
Jeff