The problem with eigenvalues and eigenvectors is that you have to factor a polynomial in order to get the eigenvalues. We can’t do this in general. All we can do is find the eigenvalues approximately. But an approximate eigenvalue is never enough to get the eigenvector because
However, there are numerical methods to do this in the case that the polynomial does not factor. I am going to mention how to get the answer using MATLAB.
To find the eigenvalues enter A and follow with ;. Then type eig(A) and press return. It will give numerical approximation of the eigenvalues. If you want to have it find the exact values, you type eig(sym(A)) and press return. To do this last thing, you need to have the symbolic math package installed.
For example, if your matirix is
You would type the following: >>A=[1,1,0;-1,0,-1;2,1,3]; and then eig(sym(A)) and return, you will get the eigenvalues 1,1,2 listed in a column. This is correct. The matrix has a repeated eigenvalue of 1. If you want to get the eigenvectors also, you would type >>A=[1,1,0;-1,0,-1;2,1,3]; and then [V,D]=eig(sym(A)) and enter or if you want numerical answers, which will sometimes be all that is available, you would type [V,D]=eig(A). It will find the matrix V such that AV = V D where D is a diagonal. In the case just considered, it will only find two columns for V because this is a defective matrix. In general, however, this would give V −1AV = D and the columns of V are the eigenvectors.