Decomposing Real Linear Transformations
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
Further, we know that the restrictions 
and 
are nilpotent transformations.
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.
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April 9, 2009 -
Posted by John Armstrong |
Algebra, Linear Algebra
The formula after “For each distinct eigenvalue” does not parse.
Thanks. Not sure when the d in lambda disappeared.
[...] Real Inner Products Now that we’ve got bilinear forms, let’s focus in on when the base field is . We’ll also add the requirement that our bilinear forms be symmetric. As we saw, a bilinear form corresponds to a linear transformation . Since is symmetric, the matrix of must itself be symmetric with respect to any basis. So let’s try to put it into a canonical form! [...]
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