Let \(V\) be a finite-dimensional vector space, and let \(T:V\to V\) be a linear operator. Assume that \(T\) has all real eigenvalues (alternatively, assume weβre working over the complex numbers). Let \(A\) be the matrix of \(T\) with respect to some standard basis \(B_0\) of \(V\text{.}\)
Our goal will be to replace the basis \(B_0\) with a basis \(B\) such that the matrix of \(T\) with respect to \(B\) is as simple as possible. (Where we agree that the "simplest" possible matrix would be diagonal.)
The question is: what do we do if there arenβt enough eigenvectors to form a basis of \(V\text{?}\) When that happens, the direct sum of all the eigenspaces will not give us all of \(V\text{.}\)
Our candidate: instead of \(E_{\lambda}(T) = \ker(T-\lambda I)\text{,}\) we use \(G_\lambda(T) = \ker((T-\lambda I)^m)\text{,}\) where \(m\) is the multiplicity of \(\lambda\text{.}\)
Recall from ExerciseΒ 5.4.2 that \(\ker(T)\) and \(\operatorname{im}(T)\) are \(T\)-invariant subspaces. Let \(p(x)\) be any polynomial, and prove that \(\ker (p(T))\) and \(\operatorname{im}(p(T))\) are also \(T\)-invariant.
Applying the result of Problem 1 to the polynomial \(p(x) = (x-\lambda)^m\) shows that \(G_\lambda(T)\) is \(T\)-invariant. It is possible to show that \(\dim G_\lambda(T)=m\) but I wonβt ask you to do that. (A proof is in the book by Nicholson if you really want to see it.)
It turns out that at some point, the null spaces stabilize. If \(\operatorname{null}(A^k)=\operatorname{null}A^{k+1}\) for some \(k\text{,}\) then \(\operatorname{null}(A^k)=\operatorname{null}(A^{k+l})\) for all \(l\geq 0\text{.}\)
For each eigenvalue found in ExerciseΒ 5.5.2, compute the nullspace of \(A-\lambda I\text{,}\)\((A-\lambda I)^2\text{,}\)\((A-\lambda I)^3\text{,}\) etc. until you find two consecutive nullspaces that are the same.
By ExerciseΒ 5.5.4, any vector in \(\operatorname{null}(A-\lambda I)^m\) will also be a vector in \(\operatorname{null}(A-\lambda I)^{m+1}\text{.}\) In particular, at each step, we can find a basis for \(\operatorname{null}(A-\lambda I)^m\) that includes the basis for \(\operatorname{null}(A-\lambda I)^{m-1}\text{.}\)
For each eigenvalue found in ExerciseΒ 5.5.2, determine such a basis for the corresponding generalized eigenspace. You will want to list your vectors so that the vectors from the basis of the nullspace for \(A-\lambda I\) come first, then the vectors for the basis of the nullspace for \((A-\lambda I)^2\text{,}\) and so on.
Finally, letβs see how all of this works. Let \(P\) be the matrix whose columns consist of the vectors found in Problem 4. What do you get when you compute the matrix \(P^{-1}AP\text{?}\)