Finally, we come to the analogue of Jordan normal form over the real numbers.
Given a linear transformation on a real vector space of dimension , we can find its characteristic polynomial. We can factor a real polynomial into the product of linear terms and irreducible quadratic terms with . These give us a list of eigenvalues and eigenpairs for .
For each distinct eigenvalue we get a subspace of generalized eigenvectors, with distinct eigenvalues in total. Similarly, for each distinct eigenpair we get a subspace of generalized eigenvectors, with distinct eigenpairs in total.
We know that these subspaces are mutually disjoint. We also know that the dimension of is equal to the multiplicity of , which is the number of factors of in the characteristic polynomial. Similarly, the dimension of is twice the multiplicity of , which is the number of factors of in the characteristic polynomial. Since each linear factor contributes to the degree of the polynomial, while each irreducible quadratic contributes , we can see that the sum of the dimensions of the and is equal to the degree of the characteristic polynomial, which is the dimension of itself.
That is, we have a decomposition of as a direct sum of invariant subspaces
I’ll leave it to you to work out what this last property implies for the matrix on the generalized eigenspace of an eigenpair, in analogy with a Jordan block for an eigenvalue.