obviously x1 and x2 must equal zero so we can choose the eigenvector to be of the form:
for example we can choose the standard base vector
I´m having some trouble with calculating the eigenvectors for the following matrix:
the char. polynomial can easily be calculated to:
hence, 3 is an eigenvalue! But trying to determine the associated eigenvector I got stuck:
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.
Can someone help please help me finding this eigenvector?
thanks a lot in advance!
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: , where D is the diagonal matrix to A, and I also know that S has the eigenvectors of A in its columns, but I`m not sure how exactly to use this to determine S^-1
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.Does it matter which eigenvector comes first in S?
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:I`m not sure how exactly to use this to determine S^-1
Invertible matrix - Wikipedia, the free encyclopedia