Jordan normal form
Encyclopedia
In linear algebra
, a Jordan normal form (often called Jordan canonical form)
of a linear operator on a finite-dimensional 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 non-diagonal entries that are non-zero 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 number
s. 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
.
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 − 4In
) 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.
where each block Ji is a square matrix of the form
So there exists an invertible matrix P such that P-1AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each Ji is called a Jordan block of A. In a given Jordan block, every entry on the super-diagonal is 1.
Assuming this result, we can deduce the following properties:
Let P have column vectors pi, i = 1, ..., 4, then
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a Jordan normal form (often called Jordan canonical form)
of a linear operator on a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
is an upper triangular matrix of a particular form called Jordan matrix
Jordan matrix
In the mathematical discipline of matrix theory, a Jordan block over a ring R is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element \lambda\in R, and for the superdiagonal, which is composed of ones...
, representing the operator on some basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
. The form is characterized by the condition that any non-diagonal entries that are non-zero 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
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, then a basis on which the matrix has the required form exists if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
all eigenvalues of M lie in K, or equivalently if the characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. 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
Normal matrix
A complex square matrix A is a normal matrix ifA^*A=AA^* \ where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.If A is a real matrix, then A*=AT...
, is a special case of the Jordan normal form.
The Jordan normal form is named after Camille Jordan
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
.
Motivation
An n × n matrix A is diagonalizableDiagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
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 − 4In
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
) 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 matrixwhere each block Ji is a square matrix of the form
So there exists an invertible matrix P such that P-1AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each Ji is called a Jordan block of A. In a given Jordan block, every entry on the super-diagonal 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 λi, its geometric multiplicity is the dimension of Ker(A − λi IIdentity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
), and it is the number of Jordan blocks corresponding to λi.
- The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λi is its algebraic multiplicity.
- 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 matrixNilpotent matrixIn linear algebra, a nilpotent matrix is a square matrix N such thatN^k = 0\,for some positive integer k. The smallest such k is sometimes called the degree of N....
defined as Nij = δi,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 closed-form 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−1AP = J, i.e.Let P have column vectors pi, i = 1, ..., 4, then
-
We see that
For i = 1,2,3 we have
, i.e. pi is an eigenvector of A corresponding to the eigenvalue λi. For i=4, multiplying both sides by 
gives
But
, so
Thus,
.
Vectors such as
are called generalized eigenvectorGeneralized eigenvectorIn linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue λ is greater than its geometric multiplicity...
s of A.
Thus, given an eigenvalue λ, its corresponding Jordan block gives rise to a Jordan chain. The generator, or lead vector, say pr, of the chain is a generalized eigenvector such that (A − λ I)rpr = 0, where r is the size of the Jordan block. The vector p1 = (A − λ I)r−1pr is an eigenvector corresponding to λ. In general, pi is a preimage of pi−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 subspaceInvariant subspaceIn mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
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 {p1, ..., pr} 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 {p1, ..., pr} must contain s vectors, say {pr−s+1, ..., pr}, 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 qi be such that
Clearly qi does not lie in Ker(A − λ I) for all i. Furthermore, qi cannot be in Ran(A − λ I), for that would contradict the assumption that each pi is a lead vector in a Jordan chain. The set {qi}, being preimages of the linearly independent set {pi} under A − λ I, is also linearly independent.
Finally, we can pick any linearly independent set {z1, ..., zt} that spans
By construction, the union the three sets {p1, ..., pr}, {qr−s +1, ..., qr}, and {z1, ..., zt} 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 k1 such that
is the size of the largest Jordan block in the Jordan form of A. (This number k1 is also called the index of λ. See discussion in a following section.) The rank of
is the number of Jordan blocks of size k1. Similarly, the rank of
is twice the number of Jordan blocks of size k1 plus the number of Jordan blocks of size k1 − 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 J1 and J2 be two Jordan normal forms of A. Then J1 and J2 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 J1 and J2. This proves the uniqueness part of the statement.
Real matrices
If A is a real matrix, its Jordan form can still be non-real, however there exists a real invertible matrix P such that P-1AP = 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
is real), or is a block matrix itself, consisting of 2×2 blocks as follows (for non-real eigenvalue 
). The diagonal blocks are identical, of the form
and describe multiplication by
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 conjugateComplex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
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 calculusFunctional calculusIn mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...
: Let A be an n × n matrix with eigenvalues λ1, ..., λn, then for any polynomial p, p(A) has eigenvalues p(λ1), ..., p(λn).
Cayley–Hamilton theorem
The Cayley–Hamilton theoremCayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
asserts that every matrix A satisfies its characteristic equation: if is the characteristic polynomialCharacteristic polynomialIn linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
of , then . This can be shown via direct calculation in the Jordan form, since any Jordan block for is annihilated by where is the multiplicity of the root of , the sum of the sizes of the Jordan blocks for , 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 fieldSplitting fieldIn abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
of ; this field extension does not change the matrix 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 domainPrincipal ideal domainIn abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
of polynomials with complex coefficients. The monic element that generates I is precisely m.
Let λ1, ..., λq be the distinct eigenvalues of A, and si be the size of the largest Jordan block corresponding to λi. It is clear from the Jordan normal form that the minimal polynomial of A has degree ∑si.
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 subspaceInvariant subspaceIn mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
s of A. Every Jordan block Ji corresponds to an invariant subspace Xi. Symbolically, we put
where each Xi 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 λi, the size of its largest corresponding Jordan block si is called the index of λi and denoted by ν(λi). (Therefore the degree of the minimal polynomial is the sum of all indices.) Define a subspace Yi 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 Yi and along all the other Yj ( j ≠ i ) is called the spectral projection of A at λi and is usually denoted by P(λi ; A). Spectral projections are mutually orthogonal in the sense that P(λi ; A) P(λj ; A) = 0 if i ≠ j. Also they commute with A and their sum is the identity matrix. Replacing every λi in the Jordan matrix J by one and zeroising all other entries gives P(λi ; J), moreover if U J U -1 is the similarity transformation such that A = U J U -1 then P(λi ; A) = U P(λi ; J) U -1.
Comparing the two decompositions, notice that, in general, l ≤ k. When A is normal, the subspaces Xi's in the first decomposition are one-dimensional and mutually orthogonal. This is the spectral theoremSpectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
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 non-negative 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.
Matrices with entries in a field
Jordan reduction can be extended to any square matrix M whose entries lie in a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K. The result states that any M can be written as a sum D + N where D is semisimpleSemisimpleIn mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way...
, N is nilpotentNilpotent matrixIn linear algebra, a nilpotent matrix is a square matrix N such thatN^k = 0\,for some positive integer k. The smallest such k is sometimes called the degree of N....
, 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 sumDirect sumIn mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
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]-moduleModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
by regarding the action of x on V as application of M and extending by K-linearity. 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 ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
K[x] of the structure theorem for finitely generated modules over a principal ideal domainStructure theorem for finitely generated modules over a principal ideal domainIn mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in...
, of which it is a corollary.
Compact operators
In a different direction, for compact operatorCompact operatorIn functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
s on a Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, 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
Let X be a Banach space, L(X) be the bounded operators on X, and σ(T) denote the spectrumSpectrum (functional analysis)In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the...
of T ∈ L(X). The holomorphic functional calculusHolomorphic functional calculusIn mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ƒ of a complex argument z and an operator T, the aim is to construct an operatorf\,...
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 = {γi} 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) = {λi} is a finite discrete set in the complex plane. Let ei be the function that is 1 in some open neighborhood of λi and 0 elsewhere. By property 3 of the functional calculus, the operator
is a projection. Moreoever, let νi be the index of λi 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)ei(T) is the zero matrix.
By property 3, f(T) ei(T) = ei(T) f(T). So ei(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 ei(T) is the projection onto the subspace spanned by the Jordan chains corresponding to λi.
This explicit identification of the operators ei(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 λi, we have chosen the Taylor series expansion of f centered at λi.
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 RT 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 Bε(λ) and σ(T) is {λ}. The resolvent function RT is holomorphic on A.
Extending a result from classical function theory, RT has a Laurent seriesLaurent seriesIn mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
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 RT 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 polynomialCharacteristic polynomialIn linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
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 one-dimensional (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−1AP = J is formed by putting these vectors next to each other as follows
A computation shows that the equation P−1AP = 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 conditioningCondition numberIn the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...
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 analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
; the stable Schur decompositionSchur decompositionIn the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.- Statement :...
is often a better alternative.
Powers
If n is a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
, the nth power of a matrix in Jordan normal form will be a direct sum of upper triangular matrices, as a result of block multiplicationBlock matrixIn the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
. More specifically, after exponentiation each Jordan block will be an upper triangular block.
For example,
Further, each triangular block will consist of λn on the main diagonal,
times λn-1 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
- Frobenius normal formFrobenius normal formIn linear algebra, the Frobenius normal form, Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for matrices that reflects the structure of the minimal polynomial of A and provides a means of detecting whether another matrix B is similar to A...
- Matrix decompositionMatrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
- Canonical formCanonical formGenerally, in mathematics, a canonical form of an object is a standard way of presenting that object....
- Jordan matrixJordan matrixIn the mathematical discipline of matrix theory, a Jordan block over a ring R is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element \lambda\in R, and for the superdiagonal, which is composed of ones...
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