 Jordan normal form

In linear algebra, a Jordan normal form (often called Jordan canonical form)^{[1]} of a linear operator on a finitedimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis. The form is characterized by the condition that any nondiagonal entries that are nonzero must be equal to 1, be immediately above the main diagonal (on the superdiagonal), and have identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis on which the matrix has the required form exists if and only if all eigenvalues of M lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.
If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size. The Jordan–Chevalley decomposition is particularly simple on a basis on which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.
The Jordan normal form is named after Camille Jordan.
Contents
Motivation
An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable. Consider the following matrix:
Including multiplicity, the eigenvalues of A are λ = 1, 2, 4, 4. The dimension of the kernel of (A − 4I_{n}) is 1 (and not 2), so A is not diagonalizable. However, there is an invertible matrix P such that A = PJP^{−1}, where
The matrix J is almost diagonal. This is the Jordan normal form of A. The section Example below fills in the details of the computation.
Complex matrices
In general, a square complex matrix A is similar to a block diagonal matrix
where each block J_{i} is a square matrix of the form
So there exists an invertible matrix P such that P^{1}AP = J is such that the only nonzero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each J_{i} is called a Jordan block of A. In a given Jordan block, every entry on the superdiagonal is 1.
Assuming this result, we can deduce the following properties:
 Counting multiplicity, the eigenvalues of J, therefore A, are the diagonal entries.
 Given an eigenvalue λ_{i}, its geometric multiplicity is the dimension of Ker(A − λ_{i} I), and it is the number of Jordan blocks corresponding to λ_{i}.^{[2]}
 The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λ_{i} is its algebraic multiplicity.^{[2]}
 A is diagonalizable if and only if, for every eigenvalue λ of A, its geometric and algebraic multiplicities coincide.
 The Jordan block corresponding to λ is of the form λ I + N, where N is a nilpotent matrix defined as N_{ij} = δ_{i,j−1} (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closedform expression for the exponential exp(A).
Generalized eigenvectors
Consider the matrix A from the example in the previous section. The Jordan normal form is obtained by some similarity transformation P^{−1}AP = J, i.e.
Let P have column vectors p_{i}, i = 1, ..., 4, then
We see that
For i = 1,2,3 we have , i.e. p_{i} is an eigenvector of A corresponding to the eigenvalue λ_{i}. For i=4, multiplying both sides by (A − 4I) gives
But (A − 4I)p_{3} = 0, so
Thus, .
Vectors such as p_{4} are called generalized eigenvectors of A.
Thus, given an eigenvalue λ, its corresponding Jordan block gives rise to a Jordan chain. The generator, or lead vector, say p_{r}, of the chain is a generalized eigenvector such that (A − λ I)^{r}p_{r} = 0, where r is the size of the Jordan block. The vector p_{1} = (A − λ I)^{r−1}p_{r} is an eigenvector corresponding to λ. In general, p_{i} is a preimage of p_{i−1} under A − λ I. So the lead vector generates the chain via multiplication by (A − λ I).
Therefore, the statement that every square matrix A can be put in Jordan normal form is equivalent to the claim that there exists a basis consisting only of eigenvectors and generalized eigenvectors of A.
A proof
We give a proof by induction. The 1 × 1 case is trivial. Let A be an n × n matrix. Take any eigenvalue λ of A. The range of A − λ I, denoted by Ran(A − λ I), is an invariant subspace of A. Also, since λ is an eigenvalue of A, the dimension Ran(A − λ I), r, is strictly less than n. Let A' denote the restriction of A to Ran(A − λ I), By inductive hypothesis, there exists a basis {p_{1}, ..., p_{r}} such that A' , expressed in terms of this basis, is in Jordan normal form.
Next consider the subspace Ker(A − λ I). If
the desired result follows immediately from the rank–nullity theorem. This would be the case, for example, if A was Hermitian.
Otherwise, if
let the dimension of Q be s ≤ r. Each vector in Q is an eigenvector of A' corresponding to eigenvalue λ. So the Jordan form of A' must contain s Jordan chains corresponding to s linearly independent eigenvectors. So the basis {p_{1}, ..., p_{r}} must contain s vectors, say {p_{r−s+1}, ..., p_{r}}, that are lead vectors in these Jordan chains from the Jordan normal form of A'. We can "extend the chains" by taking the preimages of these lead vectors. (This is the key step of argument; in general, generalized eigenvectors need not lie in Ran(A − λ I).) Let q_{i} be such that
Clearly q_{i} does not lie in Ker(A − λ I) for all i. Furthermore, q_{i} cannot be in Ran(A − λ I), for that would contradict the assumption that each p_{i} is a lead vector in a Jordan chain. The set {q_{i}}, being preimages of the linearly independent set {p_{i}} under A − λ I, is also linearly independent.
Finally, we can pick any linearly independent set {z_{1}, ..., z_{t}} that spans
By construction, the union the three sets {p_{1}, ..., p_{r}}, {q_{r−s +1}, ..., q_{r}}, and {z_{1}, ..., z_{t}} is linearly independent. Each vector in the union is either an eigenvector or a generalized eigenvector of A. Finally, by rank–nullity theorem, the cardinality of the union is n. In other words, we have found a basis that consists of eigenvectors and generalized eigenvectors of A, and this shows A can be put in Jordan normal form.
Uniqueness
It can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks.
Knowing the algebraic and geometric multiplicities of the eigenvalues is not sufficient to determine the Jordan normal form of A. Assuming the algebraic multiplicity m(λ) of an eigenvalue λ is known, the structure of the Jordan form can be ascertained by analysing the ranks of the powers (A − λ I)^{m(λ)}. To see this, suppose an n × n matrix A has only one eigenvalue λ. So m(λ) = n. The smallest integer k_{1} such that
is the size of the largest Jordan block in the Jordan form of A. (This number k_{1} is also called the index of λ. See discussion in a following section.) The rank of
is the number of Jordan blocks of size k_{1}. Similarly, the rank of
is twice the number of Jordan blocks of size k_{1} plus the number of Jordan blocks of size k_{1} − 1. Iterating in this way gives the precise Jordan structure of A. The general case is similar.
This can be used to show the uniqueness of the Jordan form. Let J_{1} and J_{2} be two Jordan normal forms of A. Then J_{1} and J_{2} are similar and have the same spectrum, including algebraic multiplicities of the eigenvalues. The procedure outlined in the previous paragraph can be used to determine the structure of these matrices. Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J_{1} and J_{2}. This proves the uniqueness part of the statement.
Real matrices
If A is a real matrix, its Jordan form can still be nonreal, however there exists a real invertible matrix P such that P^{1}AP = J is a real block diagonal matrix with each block being a real Jordan block. A real Jordan block is either identical to a complex Jordan block (if the corresponding eigenvalue λ_{i} is real), or is a block matrix itself, consisting of 2×2 blocks as follows (for nonreal eigenvalue λ_{i} = a_{i} + ib_{i}). The diagonal blocks are identical, of the form
and describe multiplication by λ_{i} in the complex plane. The superdiagonal blocks are 2×2 identity matrices. The full real Jordan block is given by
This real Jordan form is a consequence of the complex Jordan form. For a real matrix the non real eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs. Taking the real and imaginary part (linear combination of the vector and its conjugate), the matrix has this form in the new basis.
Consequences
One can see that the Jordan normal form is essentially a classification result for square matrices, and as such several important results from linear algebra can be viewed as its consequences.
Spectral mapping theorem
Using the Jordan normal form, direct calculation gives a spectral mapping theorem for the polynomial functional calculus: Let A be an n × n matrix with eigenvalues λ_{1}, ..., λ_{n}, then for any polynomial p, p(A) has eigenvalues p(λ_{1}), ..., p(λ_{n}).
Cayley–Hamilton theorem
The Cayley–Hamilton theorem asserts that every matrix A satisfies its characteristic equation: if p is the characteristic polynomial of A, then p(A) = 0. This can be shown via direct calculation in the Jordan form, since any Jordan block for λ is annihilated by (X − λ)^{m} where m is the multiplicity of the root λ of p, the sum of the sizes of the Jordan blocks for λ, and therefore no less than the size of the block in question. The Jordan form can be assumed to exist over a field extending the base field of the matrix, for instance over the splitting field of p; this field extension does not change the matrix p(A) in any way.
Minimal polynomial
The minimal polynomial of a square matrix A is the unique monic polynomial of least degree, m, such that m(A) = 0. Alternatively, the set of polynomials that annihilate a given A form an ideal I in C[x], the principal ideal domain of polynomials with complex coefficients. The monic element that generates I is precisely m.
Let λ_{1}, ..., λ_{q} be the distinct eigenvalues of A, and s_{i} be the size of the largest Jordan block corresponding to λ_{i}. It is clear from the Jordan normal form that the minimal polynomial of A has degree ∑s_{i}.
While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notion of elementary divisors. The elementary divisors of a square matrix A are the characteristic polynomials of its Jordan blocks. The factors of the minimal polynomial m are the elementary divisors of the largest degree corresponding to distinct eigenvalues.
The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of the corresponding invariant subspace. If all elementary divisors are linear, A is diagonalizable.
Invariant subspace decompositions
The Jordan form of a n × n matrix A is block diagonal, and therefore gives a decomposition of the n dimensional Euclidean space into invariant subspaces of A. Every Jordan block J_{i} corresponds to an invariant subspace X_{i}. Symbolically, we put
where each X_{i} is the span of the corresponding Jordan chain, and k is the number of Jordan chains.
One can also obtain a slightly different decomposition via the Jordan form. Given an eigenvalue λ_{i}, the size of its largest corresponding Jordan block s_{i} is called the index of λ_{i} and denoted by ν(λ_{i}). (Therefore the degree of the minimal polynomial is the sum of all indices.) Define a subspace Y_{i} by
This gives the decomposition
where l is the number of distinct eigenvalues of A. Intuitively, we glob together the Jordan block invariant subspaces corresponding to the same eigenvalue. In the extreme case where A is a multiple of the identity matrix we have k = n and l = 1.
The projection onto Y_{i} and along all the other Y_{j} ( j ≠ i ) is called the spectral projection of A at λ_{i} and is usually denoted by P(λ_{i} ; A). Spectral projections are mutually orthogonal in the sense that P(λ_{i} ; A) P(λ_{j} ; A) = 0 if i ≠ j. Also they commute with A and their sum is the identity matrix. Replacing every λ_{i} in the Jordan matrix J by one and zeroising all other entries gives P(λ_{i} ; J), moreover if U J U ^{1} is the similarity transformation such that A = U J U ^{1} then P(λ_{i} ; A) = U P(λ_{i} ; J) U ^{1}.
Comparing the two decompositions, notice that, in general, l ≤ k. When A is normal, the subspaces X_{i}'s in the first decomposition are onedimensional and mutually orthogonal. This is the spectral theorem for normal operators. The second decomposition generalizes more easily for general compact operators on Banach spaces.
It might be of interest here to note some properties of the index, ν(λ). More generally, for a complex number λ, its index can be defined as the least nonnegative integer ν(λ) such that
So ν(λ) > 0 if and only if λ is an eigenvalue of A. In the finite dimensional case, ν(λ) ≤ the algebraic multiplicity of λ.
Generalizations
Matrices with entries in a field
Jordan reduction can be extended to any square matrix M whose entries lie in a field K. The result states that any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND. This is called the Jordan–Chevalley decomposition. Whenever K contains the eigenvalues of M, in particular when K is algebraically closed, the normal form can be expressed explicitly as the direct sum of Jordan blocks.
Similar to the case when K is the complex numbers, knowing the dimensions of the kernels of (M − λI)^{k} for 1 ≤ k ≤ m, where m is the algebraic multiplicity of the eigenvalue λ, allows one to determine the Jordan form of M. We may view the underlying vector space V as a K[x]module by regarding the action of x on V as application of M and extending by Klinearity. Then the polynomials (x − λ)^{k} are the elementary divisors of M, and the Jordan normal form is concerned with representing M in terms of blocks associated to the elementary divisors.
The proof of the Jordan normal form is usually carried out as an application to the ring K[x] of the structure theorem for finitely generated modules over a principal ideal domain, of which it is a corollary.
Compact operators
In a different direction, for compact operators on a Banach space, a result analogous to the Jordan normal form holds. One restricts to compact operators because every point x in the spectrum of a compact operator T, the only exception being when x is the limit point of the spectrum, is an eigenvalue. This is not true for bounded operators in general. To give some idea of this generalization, we first reformulate the Jordan decomposition in the language of functional analysis.
Holomorphic functional calculus
For more details on this topic, see holomorphic functional calculus.Let X be a Banach space, L(X) be the bounded operators on X, and σ(T) denote the spectrum of T ∈ L(X). The holomorphic functional calculus is defined as follows:
Fix a bounded operator T. Consider the family Hol(T) of complex functions that is holomorphic on some open set G containing σ(T). Let Γ = {γ_{i}} be a finite collection of Jordan curves such that σ(T) lies in the inside of Γ, we define f(T) by
The open set G could vary with f and need not be connected. The integral is defined as the limit of the Riemann sums, as in the scalar case. Although the integral makes sense for continuous f, we restrict to holomorphic functions to apply the machinery from classical function theory (e.g. the Cauchy integral formula). The assumption that σ(T) lie in the inside of Γ ensures f(T) is well defined; it does not depend on the choice of Γ. The functional calculus is the mapping Φ from Hol(T) to L(X) given by
We will require the following properties of this functional calculus:
 Φ extends the polynomial functional calculus.
 The spectral mapping theorem holds: σ(f(T)) = f(σ(T)).
 Φ is an algebra homomorphism.
The finite dimensional case
In the finite dimensional case, σ(T) = {λ_{i}} is a finite discrete set in the complex plane. Let e_{i} be the function that is 1 in some open neighborhood of λ_{i} and 0 elsewhere. By property 3 of the functional calculus, the operator
is a projection. Moreoever, let ν_{i} be the index of λ_{i} and
The spectral mapping theorem tells us
has spectrum {0}. By property 1, f(T) can be directly computed in the Jordan form, and by inspection, we see that the operator f(T)e_{i}(T) is the zero matrix.
By property 3, f(T) e_{i}(T) = e_{i}(T) f(T). So e_{i}(T) is precisely the projection onto the subspace
The relation
implies
where the index i runs through the distinct eigenvalues of T. This is exactly the invariant subspace decomposition
given in a previous section. Each e_{i}(T) is the projection onto the subspace spanned by the Jordan chains corresponding to λ_{i}.
This explicit identification of the operators e_{i}(T) in turn gives an explicit form of holomorphic functional calculus for matrices:
 For all f ∈ Hol(T),
Notice that the expression of f(T) is a finite sum because, on each neighborhood of λ_{i}, we have chosen the Taylor series expansion of f centered at λ_{i}.
Poles of an operator
Let T be a bounded operator λ be an isolated point of σ(T). (As stated above, when T is compact, every point in its spectrum is an isolated point, except possibly the limit point 0.)
The point λ is called a pole of operator T with order ν if the resolvent function R_{T} defined by
has a pole of order ν at λ.
We will show that, in the finite dimensional case, the order of an eigenvalue coincides with its index. The result also holds for compact operators.
Consider the annular region A centered at the eigenvalue λ with sufficiently small radius ε such that the intersection of the open disc B_{ε}(λ) and σ(T) is {λ}. The resolvent function R_{T} is holomorphic on A. Extending a result from classical function theory, R_{T} has a Laurent series representation on A:
where
 and C is a small circle centered at λ.
By the previous discussion on the functional calculus,
 where is 1 on and 0 elsewhere.
But we have shown that the smallest positive integer m such that
 and
is precisely the index of λ, ν(λ). In other words, the function R_{T} has a pole of order ν(λ) at λ.
Example
This example shows how to calculate the Jordan normal form of a given matrix. As the next section explains, it is important to do the computation exactly instead of rounding the results.
Consider the matrix
which is mentioned in the beginning of the article.
The characteristic polynomial of A is
This shows that the eigenvalues are 1, 2, 4 and 4, according to algebraic multiplicity. The eigenspace corresponding to the eigenvalue 1 can be found by solving the equation Av = λ v. It is spanned by the column vector v = (−1, 1, 0, 0)^{T}. Similarly, the eigenspace corresponding to the eigenvalue 2 is spanned by w = (1, −1, 0, 1)^{T}. Finally, the eigenspace corresponding to the eigenvalue 4 is also onedimensional (even though this is a double eigenvalue) and is spanned by x = (1, 0, −1, 1)^{T}. So, the geometric multiplicity (i.e. dimension of the eigenspace of the given eigenvalue) of each of the three eigenvalues is one. Therefore, the two eigenvalues equal to 4 correspond to a single Jordan block, and the Jordan normal form of the matrix A is the direct sum
There are three chains. Two have length one: {v} and {w}, corresponding to the eigenvalues 1 and 2, respectively. There is one chain of length two corresponding to the eigenvalue 4. To find this chain, calculate
Pick a vector in the above span that is not in the kernel of A − 4I, e.g., y = (1,0,0,0)^{T}. Now, (A − 4I)y = x and (A − 4I)x = 0, so {y, x} is a chain of length two corresponding to the eigenvalue 4.
The transition matrix P such that P^{−1}AP = J is formed by putting these vectors next to each other as follows
A computation shows that the equation P^{−1}AP = J indeed holds.
If we had interchanged the order of which the chain vectors appeared, that is, changing the order of v, w and {x, y} together, the Jordan blocks would be interchanged. However, the Jordan forms are equivalent Jordan forms.
Numerical analysis
If the matrix A has multiple eigenvalues, or is close to a matrix with multiple eigenvalues, then its Jordan normal form is very sensitive to perturbations. Consider for instance the matrix
If ε = 0, then the Jordan normal form is simply
However, for ε ≠ 0, the Jordan normal form is
This ill conditioning makes it very hard to develop a robust numerical algorithm for the Jordan normal form, as the result depends critically on whether two eigenvalues are deemed to be equal. For this reason, the Jordan normal form is usually avoided in numerical analysis; the stable Schur decomposition is often a better alternative.^{[3]}
Powers
If n is a natural number, the n^{th} power of a matrix in Jordan normal form will be a direct sum of upper triangular matrices, as a result of block multiplication. More specifically, after exponentiation each Jordan block will be an upper triangular block.
For example,
Further, each triangular block will consist of λ^{n} on the main diagonal, times λ^{n1} on the upper diagonal, and so on. This expression is valid for negative integer powers as well if one extends the notion of the binomial coefficients .
For example,
See also
Notes
 ^ Shilov defines the term Jordan canonical form and in a footnote says that Jordan normal form is synonymous. These terms are sometimes shortened to Jordan form. (Shilov) The term Classical canonical form is also sometimes used in the sense of this article. (James & James, 1976)
 ^ ^{a} ^{b} Horn & Johnson (1985, §3.2.1)
 ^ See Golub & Van Loan (1996), §7.6.5; or Golub & Wilkinson (1976) for details.
References
 N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience, 1958.
 Daniel T. Finkbeiner II, Introduction to Matrices and Linear Transformations, Third Edition, Freeman, 1978.
 Gene H. Golub and Charles F. Van Loan, Matrix Computations (3rd ed.), Johns Hopkins University Press, Baltimore, 1996.
 Gene H. Golub and J. H. Wilkinson, Illconditioned eigensystems and the computation of the Jordan normal form, SIAM Review, vol. 18, nr. 4, pp. 578–619, 1976.
 Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 9780521386326.
 Glenn James and Robert C. James, Mathematics Dictionary, Fourth Edition, Van Nostrand Reinhold, 1976.
 Saunders MacLane and Garrett Birkhoff, Algebra, MacMillan, 1967.
 Anthony N. Michel and Charles J. Herget, Applied Algebra and Functional Analysis, Dover, 1993.
 Georgi E. Shilov, Linear Algebra, Dover, 1977.
 Jordan Canonical Form article at mathworld.wolfram.com
Categories: Linear algebra
 Matrix theory
 Matrix normal forms
 Matrix decompositions
Wikimedia Foundation. 2010.