Unformatted text preview: ————————————————————— CHAPTER 7. —— .
The characteristic equation of the coefficient matrix is
single root of multiplicity three,
, we have , with a .
The system of algebraic equations reduce to a single equation An eigenvector vector is given by
Since the last equation has two
free variables, a second linearly independent eigenvector associated with
Therefore two solutions are obtained as
x . It follows directly that x
A and x Hence the coefficient vectors must
. Rearranging the terms , we have A
A I Given an eigenvector , it follows that A A
I I . . . Note that a linear combination of two eigenvectors, associated with the same
eigenvalue, is also an eigenvector. Consider the equation A I
The augmented matrix is Using elementary row operations, we obtain It is evident that a solution exists provided
. Let . . The components of the generalized eigenvector must satisfy ____________________________________________________________...
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- Spring '08
- Linear Algebra, eigenvector