S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 3: Linear Maps and Operators

12 Jordan Normal Form, Cayley–Hamilton, and Categories

· Denis Auroux

Original Professor Auroux’s handwritten notes (PDF), behind a password for now:

Where we are

Last time: finite-dimensional over , a linear operator. Then:

  • the eigenspaces are linearly independent, but need not span (if they do, there exists a basis of eigenvectors, hence is diagonalizable);
  • to do better, we introduced the generalized eigenspaces , which also are linearly independent invariant subspaces for , and we proved: if is algebraically closed, then
  • we’ll now see: there exists a basis of such that the matrix of is a direct sum of Jordan blocks

Combining with last week’s results, we get the Jordan normal form.

Theorem 12.1 (Jordan normal form). Let be a finite-dimensional vector space over algebraically closed (e.g. ), and . Then there exists a basis of in which the matrix of is block-diagonal, with each block a Jordan block

Remark.

  • A size 1 Jordan block is , size 2 is , … is diagonalizable all the blocks have size 1.
  • The values of that appear are exactly the eigenvalues of . There may be several blocks with the same ; their direct sum is the generalized eigenspace .
  • Proof: we’ve seen (generalized eigenspaces); is nilpotent (). We’ll now see that nilpotent operators decompose into nilpotent Jordan blocks, hence the result.

Proposition 12.2. Let be nilpotent (i.e. there exists such that ). Then there exists a basis of of the form such that if and , i.e. is a direct sum of nilpotent Jordan blocks (of sizes ).

Proof. Recall . Assume this is the smallest , i.e. but .

Claim. If a subspace satisfies (), then is injective, , and .

Indeed: for all , , we have and . In particular , i.e. : injective. Also, , and .

First step: let be such that , and pick a basis of . [These will yield Jordan blocks of size !] (E.g.: start from a basis of , extend to a basis of by adding vectors , and let be their span.)

Now by the claim, are linearly independent, and their span is but independent of .

Start from a basis of , add , and complete to a basis of by adding some other vectors (if needed: we could have ). (These will yield blocks of size .)

Let . Then .

And so on: given with , take for and extend by adding vectors as needed to build . This eventually gives a basis of , and rearranging it as we get the result. ∎

Characteristic polynomial, minimal polynomial

Let be algebraically closed, , and with the generalized eigenspaces. Call

  • the multiplicity of ();
  • = the nilpotence order of , i.e. the smallest such that ; this is the size of the largest Jordan block in .

From the above: , and is diagonalizable iff all .

Definition 12.3. The characteristic polynomial of is

The usual definition, once we have defined the determinant, is Manifestly, in a basis where is triangular (or Jordan), and this is the same thing (but we can use any basis to calculate ).

The significance is: given the matrix of in any basis, we can calculate and solve for its roots = the eigenvalues; their multiplicities = the dimensions of the generalized eigenspaces. (This also works over non-algebraically closed , without any guarantee that has any roots.)

Definition 12.4. The minimal polynomial of is

Significance: on the generalized eigenspace iff , and it is invertible on the other generalized eigenspaces. So is the simplest polynomial expression in that is zero on all the ’s, hence on . Hence:

Proposition 12.5. , and for all , iff divides .

Since the nilpotence order is always , divides , so:

Theorem 12.6 (Cayley–Hamilton). .

(This is also true over non-algebraically closed , by passing to the algebraic closure.)

A word about operators on finite-dimensional -vector spaces

Let be a real vector space (of dimension ), a linear operator. Since is not algebraically closed, might not have eigenvalues, and we can’t put in triangular or Jordan form.

Yet: every real operator has an invariant subspace of dimension 1 or 2.

Approach: work over , which is algebraically closed. How do we do this?

Definition 12.7. The complexification of is , with addition and scalar multiplication

  • This is a -vector space of dimension : if is a basis of over , then is also a basis of over .
  • Given -linear, we can extend it to -linear simply by . Choosing a basis as above, the matrix of is the same as that of ().

But now… is guaranteed to have an eigenvector! (And generalized eigenspaces, and a Jordan form, …)

Let be an eigenvector of for the eigenvalue , . There are two cases:

  • If , then , so and are eigenvectors of with the same eigenvalue (if they are nonzero; one of them is). (Warning: the multiplicity of for has no reason to be even.)
  • If , then (compare real and imaginary parts!), i.e. is an eigenvector of with eigenvalue . It follows that and are linearly independent, and span a 2-dimensional invariant subspace :

(One could further study block triangular decompositions of etc. starting from .)

Notes not reached in this lecture, covered at the start of {{LINK:L14}}

Interlude: the language of categories

(Then we’ll return to (bi)linear algebra.)

Definition 12.8. A category is a collection of objects, plus for each pair of objects a collection of morphisms , and an operation called composition of morphisms, such that

  1. every object has an identity morphism such that for all , ;
  2. composition is associative: .

Example 12.9.

  1. The category of sets, = all maps .
  2. : finite-dimensional vector spaces over , linear maps.
  3. Groups, group homomorphisms.
  4. Topological spaces, continuous maps.

Definition 12.10. is an isomorphism if there exists (the inverse isomorphism) such that and .

Check:

  • the inverse of , if it exists, is unique;
  • is an isomorphism; iso iso; isos iso.

So the automorphisms of , , form a group.

  • Isomorphic objects have isomorphic automorphism groups: an isomorphism determines an isomorphism of groups , .

Example 12.11.

  1. In Sets, a finite set with elements .
  2. an -dimensional vector space over , the invertible matrices.