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Computation of the Jordan Normal Form of a Matrix

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COMPUTATION OF THE JORDAN NORMAL FORM OF A MATRIX USING VERSAL DEFORMATIONS ALEXEI A. MAILYBAEV Abstract. Numerical problem of nding multiple eigenvalues of real and complex nonsymmet- ric matrices is considered. Using versal deformation theory, explicit formulae for approximations of matrices having nonderogatory multiple eigenvalues in the vicinity of a given matrix A are de- rived. These formulae provide lo cal information on the geometry of a set of matrices having a given nonderogatory Jordan structure, distance to the nearest matrix with a multiple eigenvalue of given multiplicity,values of multiple eigenvalues, and corresp onding Jordan chains (generalized eigenvec- tors). The formulae use only eigenvalues and eigenvectors of the initial matrix A,whichtypically has only simple eigenvalues. Both the case of matrix families (matrices smo othly dep endentonseveral parameters) and the case of real or complex matrix spaces are studied. Several examples showing simplicity and eciency of the suggested metho d are given. Key words. Jordan normal form, multiple eigenvalue, generalized eigenvector, matrix family, versal deformation, bifurcation diagram AMS sub ject classi cations. 65F15, 15A18, 15A21 1. Intro duction. Computation of the Jordan normal form (JNF) of a square nonsymmetric matrix A is needed in many branches of mathematics, mechanics, and physics. Due to its great imp ortance this problem was intensively studied during the last century. It is well known that a generic (typical) matrix has only simple eigenvalues, which means that its JNF is a diagonal matrix [1,2]. The problem of nding the JNF of a generic matrix has b een well studied b oth from theoretical and practical (numerical) sides; see [24]. Nevertheless, manyinteresting and imp or- tant phenomena, asso ciated with qualitativechanges in the dynamics of mechanical systems [17,18, 22], stability optimization [3, 14, 20], and eigenvalue b ehavior under matrix p erturbations [4, 21, 23], are connected with degenerate matrices, i.e., matrices having multiple eigenvalues. In the presence of multiple eigenvalues the numerical problem of calculation of the JNF b ecomes very complicated due to its instability, since arbitrarily small p ertur- bations (caused, for example, by round-o errors) destroy the nongeneric structure. This means that we should consider a matrix given with some accuracy,thus, substi- tuting the matrix A by its neighb or in the matrix space. Suchformulation leads to several imp ortant problems. The rst problem left op en by Wilkinson [25, 26] is to nd a distance from a given matrix A to the nearest nondiagonalizable (degenerate) b matrix A. This problem can b e reformulated in the extended form: to nd a distance (with resp ect to a given norm) from a matrix A to the b nearest matrix A having a given Jordan structure; b to nd a matrix A with the least generic Jordan structure in a given neigh- b orho o d of the matrix A (the least generic Jordan structure means that a set of matrices with this Jordan structure has the highest co dimension). Here we assume that two matrices have the same Jordan structure, if their JNFs di er only byvalues of di erent eigenvalues. Co dimensions of sets of matrices having the same Jordan structure were studied by Arnold [1, 2]. According to Arnold, matrices b A with the least generic Jordan structure have the most p owerful in uence on the Institute of Mechanics, Moscow State Lomonosov University,Michurinsky pr. 1, 117192 Moscow, Russia ([email protected]). 1 2 ALEXEI A. MAILYBAEV eigenvalue b ehavior in the neighb orho o d of the matrix A. Therefore, the ab ove stated problems represent the key p oint in the numerical analysis of multiple eigenvalues. Several authors addressed the problem of nding b ounds on the distance to the nearest degenerate matrix; see [5,6,9,15,25,26] and more references therein. A nu- merical metho d for nding the JNF of a degenerate matrix given with some accuracy was prop osed in [12,13]. This metho d is based on making a numb er of successive transformations of the matrix and deleting small elements whenever appropriate. Nev- ertheless, this pro cedure do esn't provide the least generic JNF, and it is restricted to the use of the Frob enius matrix norm (for example, it can not b e used in the case, when di erententries of a matrix are given with di erent accuracy). Another metho d based on singular value decomp ositions was given in [19]. This metho d is simpler, but it has the same prop erties. More references of JNF computation metho ds can b e found in [8]. It is understo o d now that the problem of calculation of the nondiagonal JNF needs the singularity theory approach based on the use of versal deformations (lo cally generic families of matrices). Versal deformations were intro duced by Arnold [1, 2] in order to stabilize the JNF transformation, but no numerical metho ds were suggested. Fairgrieve[9] succeeded in using the idea of versality for construction of a stable numerical pro cedure, which calculates matrices of a given Jordan structure. Though this metho d do es not provide the nearest matrix of a given Jordan structure, it shows the usefulness of the singularity theory for the numerical eigenvalue problem. In [7] versal deformations were used for improvement of an existing numerical program in a similar problem of nding the Kronecker normal form of a matrix p encil. Application of the singularity theory to computation of a distance from a matrix A to the nearest matrix with a double eigenvalue [15] showed that this numerical problem is very complex even in the case of a double eigenvalue. This pap er is intended to prove that the versal deformation theory can b e an indep endentpowerful to ol for analysis of multiple eigenvalues. It is shown that us- ing versal deformations wecanavoid a numb er of transformations and singular value decomp ositions (used in most algorithms). Instead of this, explicit approximate for- mulae describing the geometry of a set of matrices with given Jordan structure in the vicinityofa given matrix A are derived. These formulae use only information on eigenvalues and eigenvectors of the matrix A. Since the matrix A is typically nondegenerate, i.e., has only simple eigenvalues, its eigenvalues and eigenvectors can b e obtained by means of standart co des. As a result, we obtain approximations of a distance from A to the nearest matrix having a given Jordan structure, its multiple eigenvalues, and the Jordan chains (generalized eigenvectors). The metho d do es not dep end on the norm used in the matrix space. Moreover, it allows analyzing multiple eigenvalues of matrices dep endent on several parameters (matrix families), where we can not vary matrix elements indep endently.Approximations derived in the pap er 2 have the accuracy O (" ), where " is the distance from A to a matrix we are lo ok- ing for. All these prop erties make the suggested approach useful and attractive for applications. In this pap er the case of a nonderogatory JNF (every eigenvalue has one corre- sp onding Jordan blo ck) is considered. Derogatory Jordan structures require further investigation and new ideas. Diculties app earing in the derogatory case are asso ci- ated with the lack of uniqueness of the transformation to a versal deformation and will b e discussed in the conclusion. The pap er is organized as follows. In section 2 some concepts of the singularity COMPUTATION OF THE JORDAN NORMAL FORM 3 theory are intro duced and the general idea of the pap er is describ ed. Sections 3 and 4 give approximate formulae for matrices having a prescrib ed nonderogatory Jordan structure in the neighb orho o d of a given generic (nondegenerate) complex or real matrix A. In section 5 the same problem is studied in the case, when the initial matrix A is degenerate. Conclusion discusses extension of the metho d to the derogatory case and p ossibilities for its application to sp eci c typ es of matrices or matrix p encils. In the pap er matrices are denoted by b old capital letters, vectors take the form of b old small letters, and scalars are represented by small italic characters. 2. Bifurcation diagram. Let us consider an m m complex (real) nonsym- metric matrix A holomorphically (smo othly) dep endentonavector of complex (real) parameters p =(p ;:::;p ). The space of all m m matrices can b e considered as a 1 n sp ecial case, where all entries of A are indep endent parameters. Matrices A and B are said to have the same Jordan structure if there exists one- to-one corresp ondence b etween di erent eigenvalues of A and B such that sizes of Jordan blo cks in the JNFs of A and B are equal for corresp onding eigenvalues (JNFs of the matrices A and B di er only byvalues of their eigenvalues). For example, among the following matrices 1 1 0 1 0 0 2 1 0 3 0 0 5 1 0 A A @ A @ @ (2.1) ; C = 0 2 0 ; B = 0 2 1 A = 0 5 0 0 0 2 0 0 2 0 0 2 the matrices A and B have the same Jordan structure, while the matrix C has a di erent Jordan structure. In the generic case a set of values of the vector p corresp onding to the matrices A(p)having the same Jordan structure represents a smo oth submanifold of the parameter space [1, 2]. This submanifold is called a stratum (also called a colored orbit ofamatrix).Abifurcation diagram of the matrix family A(p) is a partition (strati cation) of the parameter space according to the Jordan structure of the matrix A(p)[1, 2].
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