7.5.1 Generalized Kernals (Null Spaces)
We want to look at general eigenvectors: things that are not actually eigenvectors but ”kinda want to be” in a generalized sense. What in tarnation does that mean!? Simplest Eigenspace is just , so let’s focus on that to get an understanding:
Definition 7.5.1 (Generalized kernal). The generalized kernal of is . These are all the vecrtors that are evnetually sent to by repeatedly applying .
Note. We note that is not standard notation. Auroux likes it though, so that is what we are going with! If you meet a stranger on the street and bring up gker be sure to tell them what you mean, lest they be confused.
The difference between the kernal and generalized kernal can be summed up by this: you are in the kernal if you go to in one step; if you apply itterative steps——and end up at you are in the generalized kernal. If I look at the kernal of I notice that it is in the kernal of ; then too this holds for more! They may or may not get larger but they are subsets:
Proposition 7.5.1. If then there is no strictly larger kernal.
Proof. . So Since the sequence stops increasing after at most steps, . ∎
Example 7.5.1. with and then but .
Note. The generalized kernal is often referred to as the null space (this is the terminology that Axler uses, for example). I may use this interchangeably.
Since it is not the case that for every , we can use the following proposition as a substitute:
Proposition 7.5.2 (Generalized kernel-image decomposition). Suppose and let be the smallest positive integer such that . Then In particular, .
Proof. We verify the two conditions for a direct sum decomposition.
Intersection is trivial: Suppose . Write for some . Then , so .
We claim . By the stabilization property, once the kernel sequence stabilizes at step , it stays constant: . Thus , which means .
Sum is all of : By rank-nullity applied to : Since we just showed , the sum is direct, and its dimension equals . Hence it equals . ∎
Note. This decomposition is the key structural result for understanding non-diagonalizable operators. The generalized kernel captures the “nilpotent part” of , while is where acts invertibly. On , repeated application of eventually kills everything; on , the restriction of is an isomorphism.
Corollary 7.5.1 (Restriction to image is an isomorphism). With notation as above, the restriction is an isomorphism.
Proof. First, is -invariant since . For injectivity: if and , then . But , so . Since is injective on a finite-dimensional space, it’s also surjective onto its image, which equals by -invariance. ∎
7.5.2 Generalized Eigenspaces
Now we extend the idea of generalized kernels to arbitrary eigenvalues, not just .
Definition 7.5.2 (Generalized eigenspace). Let and let be an eigenvalue of . The generalized eigenspace of corresponding to is Equivalently, .
Note. The ordinary eigenspace is always contained in , but may be strictly larger. The elements of are called generalized eigenvectors—they are not true eigenvectors, but they are “eventually killed” by .
Example 7.5.2 (Generalized eigenspace of the shear). Recall the shear with matrix . The only eigenvalue is , and the eigenspace is , which is one-dimensional.
However, .
So the generalized eigenspace is . The vector is a generalized eigenvector: , but .
Proposition 7.5.3 (Basic properties of generalized eigenspaces). Let with eigenvalue . Then:
- .
- is -invariant.
- On , the operator is nilpotent.
- If are distinct eigenvalues, then .
Proof. (1) is immediate: if , then certainly for all .
(2) Let , so for some . Since commutes with , we have Thus .
(3) By definition, for some . So restricted to is the zero map.
(4) Suppose with . Then there exist such that and .
The polynomials and are coprime (they share no common roots). By Bézout’s identity, there exist polynomials such that .
Evaluating at : . Applying to : ∎
Note. Part (4) shows that generalized eigenspaces for distinct eigenvalues intersect trivially, just like ordinary eigenspaces. The proof uses the coprimality of and —this is a technique that will reappear in the generalized eigenspace decomposition.
7.5.3 The Generalized Eigenspace Decomposition
The following theorem is the main structural result for operators on finite-dimensional vector spaces over algebraically closed fields.
Theorem 7.5.1 (Generalized eigenspace decomposition). Let be a finite-dimensional vector space over an algebraically closed field , and let . Let be the distinct eigenvalues of . Then Moreover, .
Proof. We proceed in two steps.
Step 1: The sum is direct. We already showed that for . More generally, suppose where . We show each .
For each , let be such that . Define and Since is not a root of , the polynomials and are coprime. By Bézout, there exist with .
Now, for all (since divides ). Applying to : Thus .
But from and both terms being zero, we get .
Step 2: The sum equals . Define where (we’ll justify this choice shortly). We claim , i.e., annihilates .
Consider the polynomial . These polynomials are pairwise coprime (each excludes exactly the factor ). Their gcd is 1, so there exist with .
For any , write . We claim . Indeed: Once we establish that (which follows from Cayley-Hamilton, proved later, or can be verified by dimension counting), this shows is a sum of elements from the .
Alternative dimension argument: Since is algebraically closed, has at least one eigenvalue. Apply the decomposition where is the stabilization index. The first summand is . On the second summand, is invertible, so is not an eigenvalue of restricted there. Proceed by induction on . ∎
Corollary 7.5.2 (Dimension of generalized eigenspaces). With notation as above, if is the characteristic polynomial, then .
Note. The exponent in the characteristic polynomial is called the algebraic multiplicity of . The dimension of the ordinary eigenspace is called the geometric multiplicity. We always have: The operator is diagonalizable if and only if geometric multiplicity equals algebraic multiplicity for every eigenvalue.
Example 7.5.3 (Decomposition for a matrix). Consider with matrix The characteristic polynomial is , so has algebraic multiplicity 2 and has algebraic multiplicity 1.
For : So , which has dimension 2.
For : , dimension 1.
Indeed, .