# Normal form (for matrices)

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The normal form of a matrix is a matrix of a pre-assigned special form obtained from by means of transformations of a prescribed type. One distinguishes various normal forms, depending on the type of transformations in question, on the domain to which the coefficients of belong, on the form of , and, finally, on the specific nature of the problem to be solved (for example, on the desirability of extending or not extending on transition from to , on the necessity of determining from uniquely or with a certain amount of arbitrariness). Frequently, instead of "normal form" one uses the term "canonical form of a matrixcanonical form" . Among the classical normal forms are the following. (Henceforth denotes the set of all matrices of rows and columns with coefficients in .)

## The Smith normal form.

Let be either the ring of integers or the ring of polynomials in with coefficients in a field . A matrix is called equivalent to a matrix if there are invertible matrices and such that . Here is equivalent to if and only if can be obtained from by a sequence of elementary row-and-column transformations, that is, transformations of the following three types: a) permutation of the rows (or columns); b) addition to one row (or column) of another row (or column) multiplied by an element of ; or c) multiplication of a row (or column) by an invertible element of . For transformations of this kind the following propositions hold: Every matrix is equivalent to a matrix of the form

where for all ; divides for ; and if , then all are positive; if , then the leading coefficients of all polynomials are 1. This matrix is called the Smith normal form of . The are called the invariant factors of and the number is called its rank. The Smith normal form of is uniquely determined and can be found as follows. The rank of is the order of the largest non-zero minor of . Suppose that ; then among all minors of of order there is at least one non-zero. Let , , be the greatest common divisor of all non-zero minors of of order (normalized by the condition for and such that the leading coefficient of is 1 for ), and let . Then , . The invariant factors form a full set of invariants of the classes of equivalent matrices: Two matrices in are equivalent if and only if their ranks and their invariant factors with equal indices are equal.

The invariant factors split (in a unique manner, up to the order of the factors) into the product of powers of irreducible elements of (which are positive integers when , and polynomials of positive degree with leading coefficient 1 when ):

where the are non-negative integers. Every factor for which is called an elementary divisor of (over ). Every elementary divisor of occurs in the set of all elementary divisors of with multiplicity equal to the number of invariant factors having this divisor in their decompositions. In contrast to the invariant factors, the elementary divisors depend on the ring over which is considered: If , is an extension of and , then, in general, a matrix has distinct elementary divisors (but the same invariant factors), depending on whether is regarded as an element of or of . The invariant factors can be recovered from the complete collection of elementary divisors, and vice versa.

For a practical method of finding the Smith normal form see, for example, [1].

The main result on the Smith normal form was obtained for (see [7]) and (see [8]). With practically no changes, the theory of Smith normal forms goes over to the case when is any principal ideal ring (see [3], [6]). The Smith normal form has important applications; for example, the structure theory of finitely-generated modules over principal ideal rings is based on it (see [3], [6]); in particular, this holds for the theory of finitely-generated Abelian groups and theory of the Jordan normal form (see below).

## The natural normal form.

Let be a field. Two square matrices are called similar over if there is a non-singular matrix such that . There is a close link between similarity and equivalence: Two matrices are similar if and only if the matrices and , where is the identity matrix, are equivalent. Thus, for the similarity of and it is necessary and sufficient that all invariant factors, or, what is the same, the collection of elementary divisors over of and , are the same. For a practical method of finding a for similar matrices and , see [1], [4].

The matrix is called the characteristic matrix of , and the invariant factors of are called the similarity invariants of ; there are of them, say . The polynomial is the determinant of and is called the characteristic polynomial of . Suppose that and that for the degree of is greater than 1. Then is similar over to a block-diagonal matrix of the form

where for a polynomial

denotes the so-called companion matrix

The matrix is uniquely determined from and is called the first natural normal form of (see [1], [2]).

Now let be the collection of all elementary divisors of . Then is similar over to a block-diagonal matrix (cf. Block-diagonal operator) whose blocks are the companion matrices of all elementary divisors of :

The matrix is determined from only up to the order of the blocks along the main diagonal; it is called the second natural normal form of (see [1], [2]), or its Frobenius, rational or quasi-natural normal form (see [4]). In contrast to the first, the second natural form changes, generally speaking, on transition from to an extension.

## The Jordan normal form.

Let be a field, let , and let be the collection of all elementary divisors of over . Suppose that has the property that the characteristic polynomial of splits in into linear factors. (This is so, for example, if is the field of complex numbers or, more generally, any algebraically closed field.) Then every one of the polynomials has the form for some , and, accordingly, has the form . The matrix in of the form

where , , is called the hypercompanion matrix of (see [1]) or the Jordan block of order with eigenvalue . The following fundamental proposition holds: A matrix is similar over to a block-diagonal matrix whose blocks are the hypercompanion matrices of all elementary divisors of :

The matrix is determined only up to the order of the blocks along the main diagonal; it is a Jordan matrix and is called the Jordan normal form of . If does not have the property mentioned above, then cannot be brought, over , to the Jordan normal form (but it can over a finite extension of ). See [4] for information about the so-called generalized Jordan normal form, reduction to which is possible over any field .

Apart from the various normal forms for arbitrary matrices, there are also special normal forms of special matrices. Classical examples are the normal forms of symmetric and skew-symmetric matrices. Let be a field. Two matrices are called congruent (see [1]) if there is a non-singular matrix such that . Normal forms under the congruence relation have been investigated most thoroughly for the classes of symmetric and skew-symmetric matrices. Suppose that and that is skew-symmetric, that is, . Then is congruent to a uniquely determined matrix of the form

which can be regarded as the normal form of under congruence. If is symmetric, that is, , then it is congruent to a matrix of the form

where for all . The number is the rank of and is uniquely determined. The subsequent finer choice of the depends on the properties of . Thus, if is algebraically closed, one may assume that ; if is the field of real numbers, one may assume that and for a certain . is uniquely determined by these properties and can be regarded as the normal form of under congruence. See [6], [10] and Quadratic form for information about the normal forms of symmetric matrices for a number of other fields, and also about Hermitian analogues of this theory.

A common feature in the theories of normal forms considered above (and also in others) is the fact that the admissible transformations over the relevant set of matrices are determined by the action of a certain group, so that the classes of matrices that can be carried into each other by means of these transformations are the orbits (cf. Orbit) of this group, and the appropriate normal form is the result of selecting in each orbit a certain canonical representative. Thus, the classes of equivalent matrices are the orbits of the group (where is the group of invertible square matrices of order with coefficients in ), acting on by the rule , where . The classes of similar matrices are the orbits of on acting by the rule , where . The classes of congruent symmetric or skew-symmetric matrices are the orbits of the group on the set of all symmetric or skew-symmetric matrices of order , acting by the rule , where . From this point of view every normal form is a specific example of the solution of part of the general problem of orbital decomposition for the action of a certain transformation group.

#### References

 [1] M. Markus, "A survey of matrix theory and matrix inequalities" , Allyn & Bacon (1964) [2] P. Lancaster, "Theory of matrices" , Acad. Press (1969) [3] S. Lang, "Algebra" , Addison-Wesley (1974) [4] A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) [5] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) [6] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) [7] H.J.S. Smith, "On systems of linear indeterminate equations and congruences" , Collected Math. Papers , 1 , Chelsea, reprint (1979) pp. 367–409 [8] G. Frobenius, "Theorie der linearen Formen mit ganzen Coeffizienten" J. Reine Angew. Math. , 86 (1879) pp. 146–208 [9] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) [10] J.-P. Serre, "A course in arithmetic" , Springer (1973) (Translated from French)

The Smith canonical form and a canonical form related to the first natural normal form are of substantial importance in linear control and system theory [a1], [a2]. Here one studies systems of equations , , , and the similarity relation is: . A pair of matrices , is called completely controllable if the rank of the block matrix

is . Observe that , so that a canonical form can be formed by selecting independent column vectors from . This can be done in many ways. The most common one is to test the columns of for independence in the order in which they appear in . This yields the following so-called Brunovskii–Luenberger canonical form or block companion canonical form for a completely-controllable pair :

where is a matrix of size for certain , , of the form

and for is the -th standard basis vector of ; the with have arbitrary coefficients . Here the 's denote coefficients which can take any value. If or is zero, the block is empty (does not occur). Instead of any field can be used. The are called controllability indices or Kronecker indices. They are invariants.

Canonical forms are often used in (numerical) computations. This must be done with caution, because they may not depend continuously on the parameters [a3]. For example, the Jordan canonical form is not continuous; an example of this is:

The matter of continuous canonical forms has much to do with moduli problems (cf. Moduli theory). Related is the matter of canonical forms for families of objects, e.g. canonical forms for holomorphic families of matrices under similarity [a4]. For a survey of moduli-type questions in linear control theory cf. [a5].

In the case of a controllable pair with , i.e. is a vector , the matrix is cyclic, see also the section below on normal forms for operators. In this special case there is just one block (and one vector ). This canonical form for a cyclic matrix with a cyclic vector is also called the Frobenius canonical form or the companion canonical form.

#### References

 [a1] W.A. Wolovich, "Linear multivariable systems" , Springer (1974) [a2] J. Klamka, "Controllability of dynamical systems" , Kluwer (1990) [a3] S.H. Golub, J.H. Wilkinson, "Ill conditioned eigensystems and the computation of the Jordan canonical form" SIAM Rev. , 18 (1976) pp. 578–619 [a4] V.I. Arnol'd, "On matrices depending on parameters" Russ. Math. Surv. , 26 : 2 (1971) pp. 29–43 Uspekhi Mat. Nauk , 26 : 2 (1971) pp. 101–114 [a5] M. Hazewinkel, "(Fine) moduli spaces for linear systems: what are they and what are they good for" C.I. Byrnes (ed.) C.F. Martin (ed.) , Geometrical Methods for the Theory of Linear Systems , Reidel (1980) pp. 125–193 [a6] H.W. Turnball, A.C. Aitken, "An introduction to the theory of canonical matrices" , Blackie & Son (1932)

A normal form of an operator is a representation, up to an isomorphism, of a self-adjoint operator acting on a Hilbert space as an orthogonal sum of multiplication operators by the independent variable.

To begin with, suppose that is a cyclic operator; this means that there is an element such that every element has a unique representation in the form , where is a function for which

here , , is the spectral resolution of . Let be the space of square-integrable functions on with weight , and let be the multiplication operator by the independent variable, with domain of definition

Then the operators and are isomorphic, ; that is, there exists an isomorphic and isometric mapping such that and .

Suppose, next, that is an arbitrary self-adjoint operator. Then can be split into an orthogonal sum of subspaces on each of which induces a cyclic operator , so that , and . If the operator is given on , then .

The operator is called the normal form or canonical representation of . The theorem on the canonical representation extends to the case of arbitrary normal operators (cf. Normal operator).

#### References

 [1] A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian) [2] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert spaces" , 1–2 , Pitman (1981) (Translated from Russian)

V.I. Sobolev

The normal form of an operator is a representation of , acting on a Fock space constructed over a certain space , where is a measure space, in the form of a sum

 (1)

where () are operator-valued generalized functions generating families of annihilation operators and creation operators :

In each term of expression (1) all factors , , stand to the right of all factors , , and the (possibly generalized) functions in the two sets of variables , , are, in the case of a symmetric (Boson) Fock space, symmetric in the variables of each set separately, and, in the case of an anti-symmetric (Fermion) Fock space, anti-symmetric in these variables.

For any bounded operator the normal form exists and is unique.

The representation (1) can be rewritten in a form containing the annihilation and creation operators directly:

 (2)

where is an orthonormal basis in and the summation in (2) is over all pairs of finite collections , of elements of this basis.

In the case of an arbitrary (separable) Hilbert space the normal form of an operator acting on the Fock space constructed over is determined for a fixed basis in by means of the expression (2), where , , , are families of annihilation and creation operators acting on .

#### References

 [1] F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)

R.A. Minlos

#### References

 [a1] N.N. [N.N. Bogolyubov] Bogolubov, A.A. Logunov, I.T. Todorov, "Introduction to axiomatic quantum field theory" , Benjamin (1975) (Translated from Russian) [a2] G. Källen, "Quantum electrodynamics" , Springer (1972) [a3] J. Glimm, A. Jaffe, "Quantum physics, a functional integral point of view" , Springer (1981)

The normal form of a recursive function is a method for specifying an -place recursive function in the form

 (*)

where is an -place primitive recursive function, is a -place primitive recursive function and is the result of applying the least-number operator to . Kleene's normal form theorem asserts that there is a primitive recursive function such that every recursive function can be represented in the form (*) with a suitable function depending on ; that is,

The normal form theorem is one of the most important results in the theory of recursive functions.

A.A. Markov [2] obtained a characterization of those functions that can be used in the normal form theorem for the representation (*). A function can be used as function whose existence is asserted in the normal form theorem if and only if the equation has infinitely many solutions for each . Such functions are called functions of great range.

#### References

 [1] A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian) [2] A.A. Markov, "On the representation of recursive functions" Izv. Akad. Nauk SSSR Ser. Mat. , 13 : 5 (1949) pp. 417–424 (In Russian)

V.E. Plisko

#### References

 [a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) pp. 288

A normal form of a system of differential equations

 (1)

near an invariant manifold is a formal system

 (2)

that is obtained from (1) by an invertible formal change of coordinates

 (3)

in which the Taylor–Fourier series contain only resonance terms. In a particular case, normal forms occurred first in the dissertation of H. Poincaré (see [1]). By means of a normal form (2) some systems (1) can be integrated, and many can be investigated for stability and can be integrated approximately; for systems (1) a search has been made for periodic solutions and families of conditionally periodic solutions, and their bifurcation has been studied.

## Normal forms in a neighbourhood of a fixed point.

Suppose that contains a fixed point of the system (1) (that is, ), that the are analytic at it and that are the eigen values of the matrix for . Let . Then in a full neighbourhood of the system (1) has the following normal form (2): the matrix has for a normal form (for example, the Jordan normal form) and the Taylor series

 (4)

contain only resonance terms for which

 (5)

Here , , . If equation (5) has no solutions in , then the normal form (2) is linear:

Every system (1) with can be reduced in a neighbourhood of a fixed point to its normal form (2) by some formal transformation (3), where the are (possibly divergent) power series, and for .

Generally speaking, the normalizing transformation (3) and the normal form (2) (that is, the coefficients in (4)) are not uniquely determined by the original system (1). A normal form (2) preserves many properties of the system (1), such as being real, symmetric, Hamiltonian, etc. (see , [3]). If the original system contains small parameters, one can include them among the coordinates , and then . Such coordinates do not change under a normalizing transformation (see [3]).

If is the number of linearly independent solutions of equation (5), then by means of a transformation

where the are integers and , the normal form (2) is carried to a system

(see , [3]). The solution of this system reduces to a solution of the subsystem of the first equations and to quadratures. The subsystem has to be investigated in the neighbourhood of the multiple singular point , because the do not contain linear terms. This can be done by a local method (see [3]).

The following problem has been examined (see ): Under what conditions on the normal form (2) does the normalizing transformation of an analytic system (1) converge (be analytic)? Let

for those for which

Condition : .

Condition : as .

Condition is weaker than . Both are satisfied for almost-all (relative to Lebesgue measure) and are very weak arithmetic restrictions on .

In case there is also condition (for the general case, see in ): There exists a power series such that in (4), , .

If for an analytic system (1) satisfies condition and the normal form (2) satisfies condition , then there exists an analytic transformation of (1) to a certain normal form. If (2) is obtained from an analytic system and fails to satisfy either condition or condition , then there exists an analytic system (1) that has (2) as its normal form, and every transformation to a normal form diverges (is not analytic).

Thus, the problem raised above is solved for all normal forms except those for which satisfies condition , but not , while the remaining coefficients of the normal form satisfy condition . The latter is a very rigid restriction on the coefficients of a normal form, and for large it holds, generally speaking, only in degenerate cases. That is, the basic reason for divergence of a transformation to normal form is not small denominators, but degeneracy of the normal form.

But even in cases of divergence of the normalizing transformation (3) with respect to (2), one can study properties of the solutions of the system (1). For example, a real system (1) has a smooth transformation to the normal form (2) even when it is not analytic. The majority of results on smooth normalization have been obtained under the condition that all . Under this condition, with the help of a change of finite smoothness class, a system (1) can be brought to a truncated normal form

 (6)

where the are polynomials of degree (see [4]–). If in the normalizing transformation (3) all terms of degree higher than are discarded, the result is a transformation

 (7)

(the are polynomials), that takes (1) to the form

 (8)

where the are polynomials containing only resonance terms and the are convergent power series containing only terms of degree higher than . Solutions of the truncated normal form (6) are approximations for solutions of (8) and, after the transformation (7), give approximations of solutions of the original system (1). In many cases one succeeds in constructing for (6) a Lyapunov function (or Chetaev function) such that

where and are positive constants. Then is a Lyapunov (Chetaev) function for the system (8); that is, the point is stable (unstable). For example, if all , one can take , and obtain Lyapunov's theorem on stability under linear approximation (see [7]; for other examples see the survey [8]).

From the normal form (2) one can find invariant analytic sets of the system (1). In what follows it is assumed for simplicity of exposition that . From the normal form (2) one extracts the formal set

where is a free parameter. Condition is satisfied on the set . Let be the union of subspaces of the form such that the corresponding eigen values , , , are pairwise commensurable. The formal set is analytic in the system (1). From one selects the subset that is analytic in (1) if condition holds (see [3]). On the sets and lie periodic solutions and families of conditionally-periodic solutions of (1). By considering the sets and in systems with small parameters, one can study all analytic perturbations and bifurcations of such solutions (see [9]).

## Generalizations.

If a system (1) does not lead to a normal form (2) but to a system whose right-hand sides contain certain non-resonance terms, then the resulting simplification is less substantial, but can improve the quality of the transformation. Thus, the reduction to a "semi-normal form" is analytic under a weakened condition (see ). Another version is a transformation that normalizes a system (1) only on certain submanifolds (for example, on certain coordinate subspaces; see ). A combination of these approaches makes it possible to prove for (1) the existence of invariant submanifolds and of solutions of specific form (see [9]).

Suppose that a system (1) is defined and analytic in a neighbourhood of an invariant manifold of dimension that is fibred into -dimensional invariant tori. Then close to one can introduce local coordinates

such that on , is of period , ranges over a certain domain , and (1) takes the form

 (9)

where , , and is a matrix. If and is triangular with constant main diagonal , then (under a weak restriction on the small denominators) there is a formal transformation of the local coordinates that takes the system (9) to the normal form

 (10)

where , , , and .

If among the coordinates there is a small parameter, (9) can be averaged by the Krylov–Bogolyubov method of averaging (see [10]), and the averaged system is a normal form. More generally, perturbation theory can be regarded as a special case of the theory of normal forms, when one of the coordinates is a small parameter (see [11]).

Theorems on the convergence of a normalizing change, on the existence of analytic invariant sets, etc., carry over to the systems (9) and (10). Here the best studied case is when is a periodic solution, that is, , . In this case the theory of normal forms is in many respects identical with the case when is a fixed point. Poincaré suggested that one should consider a pointwise mapping of a normal section across the periods. In this context arose a theory of normal forms of pointwise mappings, which is parallel to the corresponding theory for systems (1). For other generalizations of normal forms see [3], , [12][14].

#### References

 [1] H. Poincaré, "Thèse, 1928" , Oeuvres , 1 , Gauthier-Villars (1951) pp. IL-CXXXII [2a] A.D. [A.D. Bryuno] Bruno, "Analytical form of differential equations" Trans. Moscow Math. Soc. , 25 (1971) pp. 131–288 Trudy Moskov. Mat. Obshch. , 25 (1971) pp. 119–262 [2b] A.D. [A.D. Bryuno] Bruno, "Analytical form of differential equations" Trans. Moscow Math. Soc. (1972) pp. 199–239 Trudy Moskov. Mat. Obshch. , 26 (1972) pp. 199–239 [3] A.D. Bryuno, "Local methods in nonlinear differential equations" , 1 , Springer (1989) (Translated from Russian) [4] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) [5a] V.S. Samovol, "Linearization of a system of differential equations in the neighbourhood of a singular point" Soviet Math. Dokl. , 13 (1972) pp. 1255–1259 Dokl. Akad. Nauk SSSR , 206 (1972) pp. 545–548 [5b] V.S. Samovol, "Equivalence of systems of differential equations in the neighbourhood of a singular point" Trans. Moscow Math. Soc. (2) , 44 (1982) pp. 217–237 Trudy Moskov. Mat. Obshch. , 44 (1982) pp. 213–234 [6a] G.R. Belitskii, "Equivalence and normal forms of germs of smooth mappings" Russian Math. Surveys , 33 : 1 (1978) pp. 95–155 Uspekhi Mat. Nauk. , 33 : 1 (1978) [6b] G.R. Belitskii, "Normal forms relative to a filtering action of a group" Trans. Moscow Math. Soc. , 40 (1979) pp. 3–46 Trudy Moskov. Mat. Obshch. , 40 (1979) pp. 3–46 [6c] G.R. Belitskii, "Smooth equivalence of germs of vector fields with a single zero eigenvalue or a pair of purely imaginary eigenvalues" Funct. Anal. Appl. , 20 : 4 (1986) pp. 253–259 Funkts. Anal. i Prilozen. , 20 : 4 (1986) pp. 1–8 [7] A.M. [A.M. Lyapunov] Liapunoff, "Problème général de la stabilité du mouvement" , Princeton Univ. Press (1947) (Translated from Russian) [8] A.L. Kunitsyn, A.P. Markev, "Stability in resonant cases" Itogi Nauk. i Tekhn. Ser. Obsh. Mekh. , 4 (1979) pp. 58–139 (In Russian) [9] J.N. Bibikov, "Local theory of nonlinear analytic ordinary differential equations" , Springer (1979) [10] N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi (1961) (Translated from Russian) [11] A.D. [A.D. Bryuno] Bruno, "Normal form in perturbation theory" , Proc. VIII Internat. Conf. Nonlinear Oscillations, Prague, 1978 , 1 , Academia (1979) pp. 177–182 (In Russian) [12] V.V. Kostin, Le Dinh Thuy, "Some tests of the convergence of a normalizing transformation" Dapovidi Akad. Nauk URSR Ser. A : 11 (1975) pp. 982–985 (In Russian) [13] E.J. Zehnder, "C.L. Siegel's linearization theorem in infinite dimensions" Manuscr. Math. , 23 (1978) pp. 363–371 [14] N.V. Nikolenko, "The method of Poincaré normal forms in problems of integrability of equations of evolution type" Russian Math. Surveys , 41 : 5 (1986) pp. 63–114 Uspekhi Mat. Nauk , 41 : 5 (1986) pp. 109–152

A.D. Bryuno