# Eigenvectors and eigenspaces

• Nov 2nd 2008, 09:31 AM
Marine
Eigenvectors and eigenspaces
hi there!

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

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

the char. polynomial can easily be calculated to:

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

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

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

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.

• Nov 2nd 2008, 10:16 AM
Peritus
$
\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:

$\left( {\begin{array}{*{20}c}
c \\
0 \\
0 \\

\end{array} } \right)$

for example we can choose the standard base vector

$
\left( {\begin{array}{*{20}c}
1 \\
0 \\
0 \\

\end{array} } \right)$
• Nov 2nd 2008, 10:36 AM
Marine
of course, it´s so obvious (Headbang)

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 :))
• Nov 2nd 2008, 10:50 AM
Marine
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: $S^-1AS=D$, 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?
• Nov 2nd 2008, 12:53 PM
Peritus
Quote:

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.

Quote:

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