Regularizing algorithm for mixed matrix pencils

P. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker's canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren's algorithm to square complex matrices with respect to consimilarity transformations and to pairs of m-by-n complex matrices.Comment: 7 page

Stratification theory of matrix pairs under equivalence and contragredient equivalence

We develop the theory of perturbations of matrix pencils basing on their miniversal deformations. Several applications of this theory are given. All possible Kronecker pencils that are canonical forms of pencils in an arbitrary small neighbourhood of a given pencil were described by A. Pokrzywa (Linear Algebra Appl., 1986). His proof is very abstract and unconstructive. Even more abstract proof of Pokrzywa’s theorem was given by K. Bongartz (Advances in Mathematics, 1996); he uses the representation theory of finite dimensional algebras. The main purpose of this thesis is to give a direct, constructive, and rather elementary proof of Pokrzywa’s theorem. We first show that it is sufficient to prove Pokrzywa’s theorem only for pencils that are direct sums of at most two indecomposable Kronecker pencils. Then we prove Pokrzywa’s theorem for such pencils. The latter problem is very simplified due to the following observation: it is sufficient to find Kronecker's canonical forms of only those pencils that are obtained by miniversal perturbations of a given pencil. We use miniversal deformations of matrix pencils that are given by M. I. García-Planas and V. V. Sergeichuk (Linear Algebra Appl., 1999) because their deformations have many zero entries unlike the miniversal deformations given by A. Edelman, E. Elmroth, and B. Kagstrom (SIAM J. Matrix Anal. Appl., 1997). Thus, we give not only all possible Kronecker’s canonical forms, but also the corresponding deformations of a given pencil, which is important for applications of this theory. P. Van Dooren (Linear Algebra Appl., 1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm both to square complex matrices under consimilarity transformations and to pairs of complex matrices under mixed equivalence. We describe all pairs (A, B) of m-by-n and n-by-m complex matrices for which the product CD is a versal deformation of AB, in which (C, D) is the miniversal deformation of (A, B) under contragredient equivalence given by M. I. García-Planas and V. V. Sergeichuk (Linear Algebra Appl., 1999). We find all canonical matrix pairs (A, B) under contragredient equivalence, for which the first order induced perturbations are nonzero for all nonzero miniversal deformations of (A, B). This problem arises in the theory of differential matrix equations dx= ABx. A complex matrix pencil is called structurally stable if there exists its neighbourhood in which all pencils are strictly equivalent to it. We describe all complex matrix pencils that are structurally stable. We show that there are no pairs of complex matrices that are structurally stable with respect to contragredient equivalence.Es desenvolupa la teoria de pertorbacions de feixos de matrius a partir de les seves deformacions miniversals. Es donen diverses aplicacions d'aquesta teoria. A. Pokrzywa (Linear Algebra Appl., 1986) va descriure tots els possibles feixos en la seva forma de Kronecker que són formes canòniques dels feixos que es poden trobar en un petit entorn arbitrari d'un feix prèviament determinat. La demostració que presentava és molt abstracta i no constructiva. K. Bongartz (Advances in Mathematics, 1996) va donar una demostració encara més abstracta del teorema de Pokrzywa; utilitzant resultats de la teoria de representació d'àlgebres de dimensió finita. L’objectiu principal de aquesta tesi és presentar una demostració directa, constructiva i bastant elemental del teorema de Pokrzywa. Primer, es demostra que per a provar el teorema de Pokrzywa és suficient provar-lo solament per a feixos que són sumes directes de, com màxim, dos feixos de Kronecker indescomponibles. Per a continuació, provar el teorema de Pokrzywa per aquests feixos. L’últim problema es simplifica molt degut a la següent observació: és suficient per trobar les formes canòniques de Kronecker de només aquells feixox que s’obtenen de deformacions miniversals d’un feix determinat. Utilitzem les deformacions de feixos de matrius obtingudes per MI García-Planas i VV Sergeichuk (Linear Algebra Appl., 1999) perquè les seves deformacions tenen moltes entrades nul·les, a diferència de les deformacions miniversals obtingudes per A. Edelman, E. Elmroth i B. Kagstrom (SIAM J. Matrix Anal. Appl., 1997). Per tant, no solament donem totes les formes canòniques de Kronecker possibles, sinó també les deformacions corresponents a un feix prèviament fixat, la qual cosa és important per a les aplicacions d’aquesta teoria. P. Van Dooren (Linear Algebra Appl., 1979) va construir un algoritme per calcular tots els sumands singulars de la forma canònica de Kronecker, d’un feix de matrius. El seu algoritme utilitza solament transformacions unitàries, el que millora la seva estabilitat numèrica. Estenem l’algoritme de Van Dooren tant a matrius complexes quadrades respecte transformacions de cosimilaritat com a parells de matrius complexes respecte l’equivalència mixta. Descrivim tots els parells (A, B) de matrius complexes m per n i n per m, per les quals el producte CD és una deformació versal de AB, en la que (C, D) és la deformació miniversal de (A, B) respecte l’equivalència contragredient donada per MI García-Planas y VV Sergeichuk (Linear Algebra Appl., 1999). Descrivim tots los pares de matrius canòniques (A, B) respecte l’equivalència contragredient, per les quals les pertorbacions de primer ordre induïdes són diferents de cero para totes les deformacions miniversals no nul·les d¿(A, B). Aquest problema apareix en la teoria de les equacions matricials diferencials dx = ABx. Un feix de matrius complexes es diu estructuralment estable si existeix un entorn en el que tots els feixos són equivalents a ell respecte una relació d’equivalència considerada. Descrivim tots els feixos de matrius complexes que són estructuralment estables respecte la equivalència estricta. Mostrem que no hi ha parelles de matrius complexes que són estructuralment estables respecto l’equivalència contragredient.Postprint (published version

Perturbation analysis of a matrix differential equation x˙=ABx\dot x=ABx

Two complex matrix pairs $(A,B)$ and $(A',B')$ are contragrediently equivalent if there are nonsingular $S$ and $R$ such that $(A',B')=(S^{-1}AR,R^{-1}BS)$. M.I. Garc\'{\i}a-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair $(A,B)$ for contragredient equivalence; that is, a simple normal form to which all matrix pairs $(A + \widetilde A, B+\widetilde B)$ close to $(A,B)$ can be reduced by contragredient equivalence transformations that smoothly depend on the entries of $\widetilde A$ and $\widetilde B$. Each perturbation $(\widetilde A,\widetilde B)$ of $(A,B)$ defines the first order induced perturbation $A\widetilde{B}+\widetilde{A}B$ of the matrix $AB$, which is the first order summand in the product $(A +\widetilde{A})(B+\widetilde{B}) = AB + A\widetilde{B}+\widetilde{A}B+ \widetilde A \widetilde B$. We find all canonical matrix pairs $(A,B)$, for which the first order induced perturbations $A\widetilde{B}+\widetilde{A}B$ are nonzero for all nonzero perturbations in the normal form of Garc\'{\i}a-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations $\dot x=Cx$, whose product of two matrices: $C=AB$; using the substitution $x = Sy$, one can reduce $C$ by similarity transformations $S^{-1}CS$ and $(A,B)$ by contragredient equivalence transformations $(S^{-1}AR,R^{-1}BS)$

Roth’s solvability criteria for the matrix equations AX - XB^ = C and X - AXB^ = C over the skew field of quaternions with aninvolutive automorphism q ¿ qˆ

The matrix equation AX-XB = C has a solution if and only if the matrices A C 0 B and A 0 0 B are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X - AXB = C over a field has a solution if and only if the matrices A C 0 I and I 0 0 B are simultaneously equivalent to A 0 0 I and I 0 0 B . We extend these criteria to the matrix equations AX- ^ XB = C and X - A ^ XB = C over the skew field of quaternions with a fixed involutive automorphism q ¿ ˆq.Postprint (author's final draft

Stratification theory of matrix pairs under equivalence and contragredient equivalence

We develop the theory of perturbations of matrix pencils basing on their miniversal deformations. Several applications of this theory are given. All possible Kronecker pencils that are canonical forms of pencils in an arbitrary small neighbourhood of a given pencil were described by A. Pokrzywa (Linear Algebra Appl., 1986). His proof is very abstract and unconstructive. Even more abstract proof of Pokrzywa’s theorem was given by K. Bongartz (Advances in Mathematics, 1996); he uses the representation theory of finite dimensional algebras. The main purpose of this thesis is to give a direct, constructive, and rather elementary proof of Pokrzywa’s theorem. We first show that it is sufficient to prove Pokrzywa’s theorem only for pencils that are direct sums of at most two indecomposable Kronecker pencils. Then we prove Pokrzywa’s theorem for such pencils. The latter problem is very simplified due to the following observation: it is sufficient to find Kronecker's canonical forms of only those pencils that are obtained by miniversal perturbations of a given pencil. We use miniversal deformations of matrix pencils that are given by M. I. García-Planas and V. V. Sergeichuk (Linear Algebra Appl., 1999) because their deformations have many zero entries unlike the miniversal deformations given by A. Edelman, E. Elmroth, and B. Kagstrom (SIAM J. Matrix Anal. Appl., 1997). Thus, we give not only all possible Kronecker’s canonical forms, but also the corresponding deformations of a given pencil, which is important for applications of this theory. P. Van Dooren (Linear Algebra Appl., 1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm both to square complex matrices under consimilarity transformations and to pairs of complex matrices under mixed equivalence. We describe all pairs (A, B) of m-by-n and n-by-m complex matrices for which the product CD is a versal deformation of AB, in which (C, D) is the miniversal deformation of (A, B) under contragredient equivalence given by M. I. García-Planas and V. V. Sergeichuk (Linear Algebra Appl., 1999). We find all canonical matrix pairs (A, B) under contragredient equivalence, for which the first order induced perturbations are nonzero for all nonzero miniversal deformations of (A, B). This problem arises in the theory of differential matrix equations dx= ABx. A complex matrix pencil is called structurally stable if there exists its neighbourhood in which all pencils are strictly equivalent to it. We describe all complex matrix pencils that are structurally stable. We show that there are no pairs of complex matrices that are structurally stable with respect to contragredient equivalence.Es desenvolupa la teoria de pertorbacions de feixos de matrius a partir de les seves deformacions miniversals. Es donen diverses aplicacions d'aquesta teoria. A. Pokrzywa (Linear Algebra Appl., 1986) va descriure tots els possibles feixos en la seva forma de Kronecker que són formes canòniques dels feixos que es poden trobar en un petit entorn arbitrari d'un feix prèviament determinat. La demostració que presentava és molt abstracta i no constructiva. K. Bongartz (Advances in Mathematics, 1996) va donar una demostració encara més abstracta del teorema de Pokrzywa; utilitzant resultats de la teoria de representació d'àlgebres de dimensió finita. L’objectiu principal de aquesta tesi és presentar una demostració directa, constructiva i bastant elemental del teorema de Pokrzywa. Primer, es demostra que per a provar el teorema de Pokrzywa és suficient provar-lo solament per a feixos que són sumes directes de, com màxim, dos feixos de Kronecker indescomponibles. Per a continuació, provar el teorema de Pokrzywa per aquests feixos. L’últim problema es simplifica molt degut a la següent observació: és suficient per trobar les formes canòniques de Kronecker de només aquells feixox que s’obtenen de deformacions miniversals d’un feix determinat. Utilitzem les deformacions de feixos de matrius obtingudes per MI García-Planas i VV Sergeichuk (Linear Algebra Appl., 1999) perquè les seves deformacions tenen moltes entrades nul·les, a diferència de les deformacions miniversals obtingudes per A. Edelman, E. Elmroth i B. Kagstrom (SIAM J. Matrix Anal. Appl., 1997). Per tant, no solament donem totes les formes canòniques de Kronecker possibles, sinó també les deformacions corresponents a un feix prèviament fixat, la qual cosa és important per a les aplicacions d’aquesta teoria. P. Van Dooren (Linear Algebra Appl., 1979) va construir un algoritme per calcular tots els sumands singulars de la forma canònica de Kronecker, d’un feix de matrius. El seu algoritme utilitza solament transformacions unitàries, el que millora la seva estabilitat numèrica. Estenem l’algoritme de Van Dooren tant a matrius complexes quadrades respecte transformacions de cosimilaritat com a parells de matrius complexes respecte l’equivalència mixta. Descrivim tots els parells (A, B) de matrius complexes m per n i n per m, per les quals el producte CD és una deformació versal de AB, en la que (C, D) és la deformació miniversal de (A, B) respecte l’equivalència contragredient donada per MI García-Planas y VV Sergeichuk (Linear Algebra Appl., 1999). Descrivim tots los pares de matrius canòniques (A, B) respecte l’equivalència contragredient, per les quals les pertorbacions de primer ordre induïdes són diferents de cero para totes les deformacions miniversals no nul·les d¿(A, B). Aquest problema apareix en la teoria de les equacions matricials diferencials dx = ABx. Un feix de matrius complexes es diu estructuralment estable si existeix un entorn en el que tots els feixos són equivalents a ell respecte una relació d’equivalència considerada. Descrivim tots els feixos de matrius complexes que són estructuralment estables respecte la equivalència estricta. Mostrem que no hi ha parelles de matrius complexes que són estructuralment estables respecto l’equivalència contragredient

Generalization of Roth's solvability criteria to systems of matrix equations

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. K{\aa}gstr\"om (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{\sigma_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, in which every $X^{\sigma}$ is $X$, $X^T$, or $X^*$. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations.Comment: 11 page

Wildness of the problems of classifying two-dimensional spaces of commuting linear operators and certain Lie algebras

For each two-dimensional vector space $V$ of commuting $n\times n$ matrices over a field $\mathbb F$ with at least 3 elements, we denote by $\widetilde V$ the vector space of all $(n+1)\times(n+1)$ matrices of the form $\left[\begin{smallmatrix}A&*\\0&0\end{smallmatrix}\right]$ with $A\in V$. We prove the wildness of the problem of classifying Lie algebras $\widetilde V$ with the bracket operation $[u,v]:=uv-vu$. We also prove the wildness of the problem of classifying two-dimensional vector spaces consisting of commuting linear operators on a vector space over a field.Comment: 11 page

Differentiable families of traceless matrix triples

Montse Pallàs 4 de nov. 2019 13:31 “This is a post-peer-review, pre-copyedit version of an article published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (Online). The final authenticated version is available online at: http://dx.doi.org/10.1007/s13398-019-00754-wAnalysis of spectra of families of sets of matrices verifying certain properties is not simple because phenomena as singularities and bifurcations appear. An excellent tool for the analysis can be making use of versal deformations because of the spectrum of the family coincides with the spectrum of its versal deformation. Disposing of a versal deformation is advantageous since any perturbation of an element can be described up to equivalence by its versal deformation, and it gives the possibility to calculate bifurcation diagrams of families of elements in general position. V.I. Arnold constructed versal deformations, of a differentiable family of square matrices under conjugation and his techniques have been generalised to different cases as to matrix pencils under the strict equivalence, for example. In this paper, we present versal deformations of elements of the Lie algebra consisting of triples of traceless matrices to coefficients on $\mathbb{F} =\mathbb{ C}$ or $\mathbb{R}$, which are simultaneously diagonalizable. Study families of traceless matrix triples have great interest because the Lie algebra is related to gauge fields because they appear in the Lagrangian describing the dynamics of the field, then they are associated to 1-forms that take values on a certain Lie algebra. It is also of interest to note that triples of traceless matrices have some relevance for supergravity theories. Another application is found when we must give the instanton solution of Yang-Mills field can be presented in an octonion form, and it can be represented by triples of traceless matrices.Peer ReviewedPostprint (author's final draft

Зміст та принципи економічної гармонізації інтересів промислових підприємств

The article is devoted to the problems of harmonizing the economic interests of enterprises under the conditions of destabilizing factors of the external environment. The theoretical basis of the research is the work of foreign and domestic scientists. The study is based on a systems approach, the economic theory of development based on resources, and the theory of economic harmonies. The content of the concept of "harmonization of the economic interests of the enterprise" is defined, under which it is proposed to understand the process of harmonizing the economic interests of the enterprise and the interests of other subjects of the external environment, which provides for the reduction of the imbalance in the resource provision of all market participants and enables the further commercially successful activity of the enterprise on the basis of innovative activity, corporate social responsibility, and sustainable development. The following principles of harmonizing the interests of industrial enterprises are highlighted: complexity, planning, limitations, competitiveness, environmental friendliness, balance, continuity, and efficiency. The obtained results deepen scientific developments regarding the need for economic harmonization of domestic industrial enterprises. The structure of the balance sheet of economic entities in Ukraine for the period 2019–2021 is shown. The types of economic activity characterized by the dominance of the share of non-current assets in the total assets of the respective enterprises are highlighted. The need to consider the dependence of businesses on material costs when determining ways to harmonize their economic activity is emphasized. Areas of ensuring the harmonization of economic interests of economic entities are highlighted, namely: innovative cooperation; optimization preservation and focusing strategy; digitalization of business.Статтю присвячено проблематиці гармонізації економічних інтересів підприємств в умовах дії дестабілізуючих чинників зовнішнього середовища. Теоретичною основою дослідження є праці зарубіжних та вітчизняних учених. Дослідження базується на системному підході, економічній теорії розвитку, заснованому на ресурсах, а також на теорії економічних гармоній. Визначено зміст поняття «гармонізація економічних інтересів підприємства», під яким запропоновано розуміти процес узгодження економічних інтересів підприємства та інтересів інших суб’єктів зовнішнього середовища, що передбачає зменшення дисбалансу ресурсного забезпечення всіх учасників ринку та уможливлює подальшу комерційно успішну діяльність підприємства на основі інноваційної активності, корпоративної соціальної відповідальності та сталого розвитку. Виділено наступні принципи гармонізації інтер- есів промислових підприємств: комплексності, планування, обмеженості, конкурентоспроможності, екологічності, збалансованості, безперервності, ефективності. Отримані результати поглиблюють наукові розробки щодо необ- хідності економічної гармонізації вітчизняних промислових підприємствах. Показано структуру балансу суб’єктів господарської діяльності в Україні за період 2019–2021 рр. та виділено види економічної діяльності, котрі характеризуються домінуванням частки необоротних активів в загальному обсязі активів відповідних підприємств. Підкреслено необхідність врахування залежності бізнесу від матеріальних витрат при визначенні шляхів забезпечення гармонізації їх економічної діяльності. Виділено напрями забезпечення гармонізації економічних інтересів суб’єктів господарювання, а саме: інноваційної кооперації; оптимізаційного консервування та стратегії фокусування; діджиталізації бізнесу