How to find the generalized eigen vectors and canonical basis for the following matrix A?
\begin{bmatrix} 0 & 4 & 2\\ -3 & 8 & 3\\ 4&-8& -2 \end{bmatrix}
The eigen value is 2 with algebraic multiplicity 3 and geometric multiplicity is 2. How to find generalized eigen vector in this case? I have solved problems where algebraic multiplicity 3 and geometric multiplicity is 1, but in this question I am stuck due to the geometric multiplicity which is two.
In fact Jordan Canonical Form (JCF) gives the "pseudo diagonalization" you are looking for under the form :
$$A=VJV^{-1}$$
where
$$V=\pmatrix{-2&1&1\\-3&0&0\\4&0&1}$$
(whose columns are the eigenvectors you are looking for) and
$$J=\left(\begin{array}{cc|c}2&1&0\\0&2&0\\ \hline 0&0&2 \end{array}\right)$$
(please note the $2$s on the diagonal and the two "Jordan blocks" on the diagonal, one $2 \times 2$, the other $1 \times 1$).
I have used a software (MATLAB) for obtaining this decomposition but it can be done by hand.
For a hand calculation, see this similar example : How to order the basis vectors that put a matrix in jordan canonical form