Jordan canonical form is a representation of a linear transformation over a finite-dimensional complex vector space by a particular kind of upper triangular matrix. Every such linear transformation has a unique Jordan canonical form, which has useful properties: it is easy to describe and well-suited for computations.
Less abstractly, one can speak of the Jordan canonical form of a square matrix; every square matrix is similar to a unique matrix in Jordan canonical form, since similar matrices correspond to representations of the same linear transformation with respect to different bases, by the change of basis theorem.
Jordan canonical form can be thought of as a generalization of diagonalizability to arbitrary linear transformations (or matrices); indeed, the Jordan canonical form of a diagonalizable linear transformation (or a diagonalizable matrix) is a diagonal matrix.
A Jordan block is a square matrix of the following form: for some complex number
Call a square matrix Jordan if it is a block matrix of the form where are (not necessarily distinct) complex numbers, and each is a Jordan block.
Jordan Canonical Form Theorem:
Let be a finite-dimensional complex vector space, and let be a linear transformation. There is a unique basis of unique up to ordering of the vectors in such that the matrix of with respect to is Jordan. This matrix is called the Jordan canonical form of
Equivalently, let be an matrix with complex entries. Then is similar to a Jordan matrix, called the Jordan canonical form of which is unique up to rearrangement of the Jordan blocks.
Let Find the Jordan canonical form of What are the eigenvalues and eigenvectors of Is diagonalizable?
The characteristic polynomial of is
The -eigenspace is the kernel of which is a one-dimensional subspace generated by
These eigenvalue and eigenvector computations show that is not diagonalizable. Now note that
- Note: We will see below how vectors like can be found in general.
So let then which is the Jordan canonical form of
A diagonalizable linear transformation on a finite-dimensional vector space has the property that there is an eigenbasis of that is, there is a basis of consisting of eigenvectors of The matrix of with respect to this basis is diagonal. Another way to say this is every vector in can be written uniquely as a sum of elements in each eigenspace of More concisely, where are the eigenvalues and is the corresponding eigenspace.
For the construction of the Jordan canonical form of a linear transformation (or matrix), the idea is to replace the eigenspaces in the last sentence of the above paragraph by larger subspaces called generalized eigenspaces of such that every vector in can always be written uniquely as a sum of elements in each generalized eigenspace.
Let be a linear transformation on a complex vector space, and let be a complex number. The generalized -eigenspace is the subspace of consisting of vectors such that for some positive integer Here is the identity map.
The vector is said to be a generalized eigenvector of rank if is the smallest positive integer such that is in the kernel of
Note that generalized eigenvectors of rank are precisely the eigenvectors of because if and only if
Let the matrix from the above example. The generalized eigenspace is the same as the eigenspace : it is one-dimensional, spanned by
The generalized eigenspace is not the same as In particular, is two-dimensional, spanned by and The first of these two vectors is an eigenvector, but the second is not. Note that so while the kernel of is one-dimensional, generated by the kernel of is two-dimensional, generated by and
Each eigenvalue of a linear transformation has two different concepts of multiplicity that can be associated to it. These two multiplicities are closely related to the Jordan canonical form.
Let be a finite-dimensional complex vector space. The algebraic multiplicity of an eigenvalue of a linear transformation is the exponent of in the characteristic polynomial
The geometric multiplicity of is the dimension of the eigenspace
Facts about Multiplicities:
If has a Jordan canonical form every does, but this will not be proved until later the algebraic multiplicity of an eigenvalue is the sum of the sizes of the Jordan blocks with on the diagonal. The geometric multiplicity is the number of Jordan blocks with on the diagonal.
The algebraic multiplicity of is the dimension of the generalized eigenspace while the geometric multiplicity is the dimension of the eigenspace
Either of the above statements implies the following fact: the algebraic multiplicity is always the geometric multiplicity, and equality holds for every eigenvalue if and only if is diagonalizable. This fact can also be proved without using the Jordan canonical form theorem.
The algebraic multiplicities always add up to by the fundamental theorem of algebra applied to the characteristic polynomial. Note that this is why is assumed to be a complex vector space: the characteristic polynomial needs to factor completely into linear factors.
Continuing the example with the characteristic polynomial is so the algebraic multiplicity of is and the algebraic multiplicity of is The geometric multiplicity of is automatically and the geometric multiplicity of is as well, because the -eigenspace is one-dimensional (as seen above). This corresponds to the fact that there is only one Jordan block with eigenvalue
Note that the algebraic multiplicities sum to the number of rows (and columns) of but the geometric multiplicities sum to This is a reflection of the fact that is not diagonalizable.
Let The characteristic polynomial is which has complex roots Note that has no real eigenvalues, and it is not similar over the real numbers to a Jordan matrix. However, it is similar over the complex numbers to the matrix which is a Jordan matrix--indeed is diagonalizable over the complex numbers.
The point of this example is that has no real roots, but since every monic polynomial of degree over the complex numbers splits into a product of linear factors, must have two complex roots, which in this case both have algebraic and geometric multiplicities equal to
Here are some useful facts about generalized eigenvectors:
The set of generalized eigenvectors of rank is a subspace of
Let Then the sequence of positive integers is a nondecreasing sequence. If two consecutive terms are equal, then every subsequent term is equal, and these terms are all equal to the algebraic multiplicity of
The quantity equals the number of Jordan blocks of size in the Jordan canonical form.
So the Jordan canonical form is determined by the quantities for every eigenvalue and positive integer
Put the matrix in Jordan canonical form.
It is straightforward to compute the characteristic polynomial So the algebraic multiplicity of is and hence the geometric multiplicity is as well. The algebraic multiplicity of is so we must compute the geometric multiplicity and the structure of the generalized eigenspace. This involves looking at the kernels of the powers of the matrix which is
Now is two-dimensional (e.g. by rank-nullity since the first, second, and fourth rows are clearly a basis for its row space), spanned by and So the geometric multiplicity is i.e. there are two Jordan blocks corresponding to the eigenvalue
On the other hand, which clearly has rank 2, so is three-dimensional. It contains and a third vector that spans it is
Finally, so is four-dimensional, with a fourth spanning vector
So the sequence equals There are two Jordan blocks corresponding to the eigenvalue The sizes add up to the algebraic multiplicity, which is The fact that implies that one of the blocks has size So there must be one block of size and one block of size
Putting it all together, the Jordan canonical form for is
The Jordan canonical form is convenient for computations. In particular, matrix powers and exponentials are straightforward to compute once the Jordan canonical form is known. Here is an illustrative example.
Let Find Can you find a formula for for any positive integer
The easiest way to do this problem is to convert to a similar matrix in Jordan canonical form, and then to consider powers of this matrix. The characteristic polynomial of is The kernel of is one-dimensional, generated by so there is only one Jordan block in Therefore is similar to
To solve the rest of the problem, it is necessary to determine the matrix such that By the change of basis theorem, the first column of is an eigenvector with eigenvalue so we can take And the second column of satisfies or That is, For convenience's sake, we take and which gives Note that which will be useful later.
Now so the problem essentially reduces to the computation of This technique may be familiar in other situations where is diagonal, for instance in one derivation of the formula for the Fibonacci numbers. In that case, is trivial to compute, as its entries are just the powers of the diagonal entries of
It remains to find a formula for It is convenient to write where Then because is the zero matrix for So and
This technique generalizes to arbitrary Jordan blocks, although the computations are tedious for larger blocks because more powers of the matrix are nonzero for a Jordan block of size is zero but is not