1. ## Eigenvectors and eigenspaces

hi there!

I´m having some trouble with calculating the eigenvectors for the following matrix:

$\displaystyle \left(\begin{array}{ccc}3&2&1\\0&1&2\\0&1&-1\end{array}\right)$

the char. polynomial can easily be calculated to:

$\displaystyle (3-x)((1-x)(-1-x)-2)=(3-x)(x^2-3)$

hence, 3 is an eigenvalue! But trying to determine the associated eigenvector I got stuck:

$\displaystyle \left(\begin{array}{ccc}0&2&1\\0&-2&2\\0&1&-4\end{array}\right)\left(\begin{array}{c}x_1\\x_2\ \x_3\end{array}\right)=\left(\begin{array}{c}0\\0\ \0\end{array}\right)$

There are 3 linearly independent lines, but 3 as an eigenvalue has the algebraic multiplicity 1, which requires an geometrical multiplicity (number of basis vectors for the eigenspace) less or equal to 1.

2. $\displaystyle \left(\begin{array}{ccc}0&2&1\\0&-2&2\\0&1&-4\end{array}\right)\left(\begin{array}{c}x_1\\x_2\ \x_3\end{array}\right)=\left(\begin{array}{c}0\\0\ \0\end{array}\right)$

obviously x1 and x2 must equal zero so we can choose the eigenvector to be of the form:

$\displaystyle \left( {\begin{array}{*{20}c} c \\ 0 \\ 0 \\ \end{array} } \right)$

for example we can choose the standard base vector

$\displaystyle \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 0 \\ \end{array} } \right)$

3. of course, it´s so obvious

thanks a lot , Peritus!

Moo, thats the way the prof introduced the eigenvectors, he used your way to check them (thanks also for the latex remark, Ive been learning it by myself and still dont know the tricks )

4. btw, one more question:

Is there any way to determine the inverse matrix of say 3 by 3 and 4 by 4 matrices, without using the co-factors and the Laplacian theorem with the minors?

I know the formula: $\displaystyle S^-1AS=D$, where D is the diagonal matrix to A, and I also know that S has the eigenvectors of A in its columns, but Im not sure how exactly to use this to determine S^-1

Does it matter which eigenvector comes first in S?

5. Does it matter which eigenvector comes first in S?
No, that would only change the places of the eigenvalues on the main diagonal of the D matrix.

Im not sure how exactly to use this to determine S^-1
There are many ways to determine the inverse matrix, one of them you already mentioned in your post. Here's a link that shows other methods:

Invertible matrix - Wikipedia, the free encyclopedia