every single example I ever encountered for finding eignspace basis' always was a situation where the reduced matrix had a null space of 1 or more. but what if after plugging in the eignenvalue and reducing, there are no free variables, leading to a reduced matrix of I and no vectors in the basis? is it an empty basis? is this basis an empty set? I was hoping I could find a quick answer to this, I can't seem to find it anywhere online.
such as basis for the eigenspace corresponding to eigenvalue -1 for the matrix A =
$$ \left[
\begin{array}{cc}
1&4\\
2&3
\end{array}
\right] $$
since after I plug in eigenvalue -1 to the characteristic eq. it reduces to I giving me no free variables, and no t parameters, how do I find the basis? is it an empty set basis?
Best Answer
if $\lambda$ is an eigenvalue of $A,$ then it cannot happen the row reduced matrix $A - \lambda I$ lacks at least one free variable.
the reason is if $\lambda$ is an eigenvalue of $A,$ then $det(A - \lambda I) = 0.$ this means $A - \lambda I$ is not invertible and the $rank(A) \le n-1.$ this in turn implies that null space of $A - \lambda I$ is non trivial. this guarantees at least one free variable.
this is the reason we are able to find a nonzero vector(eigenvector)$u$ associated with the eigenvalue $\lambda$ such that $A u = \lambda u, u \neq 0.$
p.s. you may not need this information but the dimension of $A - \lambda A$ is called the geometric multiplicity of $\lambda$ and tells you the number of jordan blocks associated with $\lambda.$