We have seen in Section IS:Invariant Subspaces that generalized eigenspaces are invariant subspaces that in every instance have led to a direct sum decomposition of the domain of the associated linear transformation. This allows us to create a block diagonal matrix representation (Example ISMR4, Example ISMR6). We also know from Theorem RGEN that the restriction of a linear transformation to a generalized eigenspace is almost a nilpotent linear transformation. Of course, we understand nilpotent linear transformations very well from Section NLT:Nilpotent Linear Transformations and we have carefully determined a nice matrix representation for them.

So here is the game plan for the final push. Prove that the domain of a linear transformation always decomposes into a direct sum of generalized eigenspaces. We have unravelled Theorem RGEN at Theorem MRRGE so that we can formulate the matrix representations of the restrictions on the generalized eigenspaces using our storehouse of results about nilpotent linear transformations. Arrive at a matrix representation of any linear transformation that is block diagonal with each block being a Jordan block.

Generalized Eigenspace Decomposition

In Theorem UTMR we were able to show that any linear transformation from $V$ to $V$ has an upper triangular matrix representation (Definition UTM). We will now show that we can improve on the basis yielding this representation by massaging the basis so that the matrix representation is also block diagonal. The subspaces associated with each block will be generalized eigenspaces, so the most general result will be a decomposition of the domain of a linear transformation into a direct sum of generalized eigenspaces.

Theorem GESD (Generalized Eigenspace Decomposition) Suppose that $\ltdefn{T}{V}{V}$ is a linear transformation with distinct eigenvalues $\scalarlist{\lambda}{m}$. Then

\begin{align*} V&= \geneigenspace{T}{\lambda_1}\ds \geneigenspace{T}{\lambda_2}\ds \geneigenspace{T}{\lambda_3}\ds \cdots\ds \geneigenspace{T}{\lambda_m} \end{align*}

Proof.  

Besides a nice decomposition into invariant subspaces, this proof has a bonus for us.

Theorem DGES (Dimension of Generalized Eigenspaces) Suppose $\ltdefn{T}{V}{V}$ is a linear transformation with eigenvalue $\lambda$. Then the dimension of the generalized eigenspace for $\lambda$ is the algebraic multiplicity of $\lambda$, $\dimension{\geneigenspace{T}{\lambda_i}}=\algmult{T}{\lambda_i}$.

Proof.  

Jordan Canonical Form

Now we are in a position to define what we (and others) regard as an especially nice matrix representation. The word "canonical" has at its root, the word "canon," which has various meanings. One is the set of laws established by a church council. Another is a set of writings that are authentic, important or representative. Here we take it to mean the accepted, or best, representative among a variety of choices. Every linear transformation admits a variety of representations, and we will declare one as the best. Hopefully you will agree.

Definition JCF (Jordan Canonical Form) A square matrix is in Jordan canonical form if it meets the following requirements:

  1. The matrix is block diagonal.
  2. Each block is a Jordan block.
  3. If $\rho < \lambda$ then the block $\jordan{k}{\rho}$ occupies rows with indices greater than the indices of the rows occupied by $\jordan{\ell}{\lambda}$.
  4. If $\rho=\lambda$ and $\ell < k$, then the block $\jordan{\ell}{\lambda}$ occupies rows with indices greater than the indices of the rows occupied by $\jordan{k}{\lambda}$.

Theorem JCFLT (Jordan Canonical Form for a Linear Transformation) Suppose $\ltdefn{T}{V}{V}$ is a linear transformation. Then there is a basis $B$ for $V$ such that the matrix representation of $T$ with the following properties:

  1. The matrix representation is in Jordan canonical form.
  2. If $\jordan{k}{\lambda}$ is one of the Jordan blocks, then $\lambda$ is an eigenvalue of $T$.
  3. For a fixed value of $\lambda$, the largest block of the form $\jordan{k}{\lambda}$ has size equal to the index of $\lambda$, $\indx{T}{\lambda}$.
  4. For a fixed value of $\lambda$, the number of blocks of the form $\jordan{k}{\lambda}$ is the geometric multiplicity of $\lambda$, $\geomult{T}{\lambda}$.
  5. For a fixed value of $\lambda$, the number of rows occupied by blocks of the form $\jordan{k}{\lambda}$ is the algebraic multiplicity of $\lambda$, $\algmult{T}{\lambda}$.

Proof.  

Before we do some examples of this result, notice how close Jordan canonical form is to a diagonal matrix. Or, equivalently, notice how close we have come to diagonalizing a matrix (Definition DZM). We have a matrix representation which has diagonal entries that are the eigenvalues of a matrix. Each occurs on the diagonal as many times as the algebraic multiplicity. However, when the geometric multiplicity is strictly less than the algebraic multiplicity, we have some entries in the representation just above the diagonal (the "superdiagonal"). Furthermore, we have some idea how often this happens if we know the geometric multiplicity and the index of the eigenvalue.

We now recognize just how simple a diagonalizable linear transformation really is. For each eigenvalue, the generalized eigenspace is just the regular eigenspace, and it decomposes into a direct sum of one-dimensional subspaces, each spanned by a different eigenvector chosen from a basis of eigenvectors for the eigenspace.

Some authors create matrix representations of nilpotent linear transformations where the Jordan block has the ones just below the diagonal (the "subdiagonal"). No matter, it is really the same, just different. We have also defined Jordan canonical form to place blocks for the larger eigenvalues earlier, and for blocks with the same eigenvalue, we place the bigger ones earlier. This is fairly standard, but there is no reason we couldn't order the blocks differently. It'd be the same, just different. The reason for choosing some ordering is to be assured that there is just one canonical matrix representation for each linear transformation.

Example JCF10: Jordan canonical form, size 10.  

Cayley-Hamilton Theorem

Jordan was a French mathematician who was active in the late 1800's. Cayley and Hamilton were 19th-century contemporaries of Jordan from Britain. The theorem that bears their names is perhaps one of the most celebrated in basic linear algebra. While our result applies only to vector spaces and linear transformations with scalars from the set of complex numbers, $\complexes$, the result is equally true if we restrict our scalars to the real numbers, $\real{\null}$. It says that every matrix satisfies its own characteristic polynomial.

Theorem CHT (Cayley-Hamilton Theorem) Suppose $A$ is a square matrix with characteristic polynomial $\charpoly{A}{x}$. Then $\charpoly{A}{A}=\zeromatrix$.

Proof.