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# Canonical matrices and related questions

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УДК 512.64+512.55 Розглянуто еквiвалентнiсть матриць у кiльцi та в його пiдкiльцях блочно-трикутних i блочно-дiагональних матриць, де комутативна область головних iдеалiв, та дослiджено їхнi зв’язки.Встановлено, що коли блочно-трикутнi матрицi блочно дiагоналiзовнi, тобто еквiвалентнi до своїх головних блочних дiагоналей, то вони еквiвалентнi у пiдкiльцi блочно-трикутних матриць тодi i тiльки тодi, коли їх головнi блочнi дiагоналi еквiвалентнi у пiдкiльцi блочно-дiагональних матриць, тобто їхнi вiдповiднi дiагональнi блоки еквiвалентнi. Доведено також, що якщо блочно-трикутнi матрицi i з нормальними формами Смiта еквiвалентнi до нормальних форм Смiта в пiдкiльцi , то цi блочно-трикутнi матрицi еквiвалентнi i в пiдкiльцi .
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For any matrix X let X′ denote its transpose. It is known that if A is an n-by-n matrix over a field F, then A and A′ are congruent over F, i.e., X AX′ = A′ for some X ε GLn(F). Moreover, X can be chosen so that X2 = In, where In is the identity matrix. An algorithm is constructed to compute such an X for a given matrix A. Consequently, a new and completely elementary proof of that result is obtained. As a by-product another interesting result is also established. Let G be a semisimple complex Lie group with Lie algebra g. Let g = g0 ⊕ g1 be a Ζ2-gradation such that g1 contains a Cartan subalgebra of g. Then L.V. Antonyan has shown that every G-orbit in g meets g1. It is shown that, in the case of the symplectic group, this assertion remains valid over an arbitrary field F of characteristic different from 2. An analog of that result is proved when the characteristic is 2.
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In this paper, we introduce a quasiorder (majorization) on *-algebras with respect to the complexity of description of their representations. We show that C*(F2) 21 for any finitely generated *-algebra (algebras such that B C*(F2) are called *-wild). We show that the *-algebra generated by orthogonal projections p, p1, p2, ⋯, pn (pipj = 0 for i ≠ j) is *-wild if n ≥ 2. We also prove that *-algebras generated by a pair of idempotents and an orthogonal projection, or by a pair of idempotents q1, q2 (q1q2 = q2q1 = 0), etc., are *-wild.
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Our objective in this chapter is to show off a few examples of algebras that occur naturally. After a brief orientation toward concepts and notation, the reader is introduced to group algebras, endomorphism algebras, matrix algebras, and quaternion algebras. Along the way, there is a brief digression, which contains a hint of the connection between algebraic geometry and the theory of finite dimensional algebras over a field.
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In his paper of 1991, Dokovic found a simple canonical form for skew projectors under unitary similarity. It is shown how this form can be obtained using standard algebraic procedures: a reduction to Shur form and the use of singular expansion algorithm. The positive parameters in the form can be interpreted as the tangents of the canonical angles between the right and left eigen-subspaces of the projector.
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Up to the classification of Hermitian forms a classification has been given of triplesP=(VF; U1, U2), consisting of a finite dimensional vector space V over a field of characteristic ? 2 with a symmetric, or a skew-symmetric, or Hermitian form F and two subspaces U1, U2. Two triplesP andP' are identified with each other if there exists an isometry ? : Vf ? V'f' such that ? (Ui)=Ui', i=1, 2.
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In this paper a necessary and sufficient condition is given that a partially ordered set have tame type, i.e. that it admit classification of its indecomposable representations.Bibliography: 16 titles. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
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A square matrix pencil λA - B is said to be H-selfadjoint (H-unitary) if it satisfies A* H B = B* H A (A* H A = B* H B) for some invertible Hermitian H. Attention is focused on regular pencils (i.e., det (λA - B) ≢ 0) for which A and B are both singular. Canonical forms for the relation (A, B, H) ∼ (Y-1 AX, Y-1 BX, Y* HY) are obtained in both the complex and real cases. Also, a characterization is given for those real matrices A which are H-unitary for some H, i.e., AT H A = H for some invertible, real symmetric H.
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The classification of algebras of generic degree two, and of nonzero generic discriminant is presented. In the case of zero discriminant a structural theorem is proven.
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We classify self-adjoint operators and pairs of Hermitian forms over the real quaternions by providing canonical matrix representations. In the preliminaries we discuss the Jordan canonical form theorem for quaternionic linear endomorphisms.
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In this paper we show that the same problem contains the problem of the classification of triples {Pl, P2, P3} of orthogonal projections, where Pl ± P2- We show, moreover, that classification up to unitary equivalence of pairs of bounded self-adjoint operators would permit giving a classification up to unitary equivalence of countable sets {Ak}k~__l of bounded operators in H whose norms are jointly bounded. This confirms once again the complexity of the classification of pairs of self-adjoint operators ,$and hence also of the corresponding triples of orthogonal projections. 1. Given any set of self-adjoint operators {Ak}k~=l whose norms are jointly bounded, we construct a pair of bounded self-adjoint operators ~lA~l, ~l.4k} in the space$~ -- ~e// as follows: 1
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We deal with linear operators acting in a finite dimensional complex Hilbert space. We show that there exists a simple canonical form for projectors (not necessarily orthogonal) under unitary similarity. As a consequence we obtain a simple test for unitary similarity of projectors. IfP is a projector we show thatP andP * are unitarily similar. We also determine the isomorphism type of the algebra generated by the projectorsP andP *.
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Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B d =Spec k[(B d ] is the affine algebraic scheme whoseR-points are theB ⊗k k[Bd]-module structures onR d, and Md is a canonical B⊗k k[Bd]-module supported by k[Bd]d. Further, say that an affine subscheme Ν of Bd isclass true if the functor Fgn ∶ X → Md ⊗k[B] X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[Ν] andB. If Bd contains a class-true plane for somed, then the schemes Be contain class-true subschemes of arbitrary dimensions. Otherwise, each Bd contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF L(d, i) (X); furthermore,F L(d, i) (X) is not isomorphic toF L(l, j) (Y) if(d, i) ≠ (l, j) andX ≠ 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.
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For the groupGL(m, C)xGL(n, C) acting on the space ofmxn matrices over C, we introduce a class of subgroups which we call admissible. We suggest an algorithm to reduce an arbitrary matrix to a normal form with respect to an action of any admissible group. This algorithm covers various classification problems, including the wild problem of bringing a pair of matrices to normal form by simultaneous similarity. The classical left, right, two-sided and similarity transformations turns out to be admissible. However, the stabilizers of known normal forms (Smith's, Jordan's), generally speaking, are not admissible, and this obstructs inductive steps of our algorithm. This is the reason that we introduce modified normal forms for classical actions.
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We prove that every finitely represented vectroid is determined, up to an isomorphism, by its completed biordered set. Elementary and multielementary representations of such vectroids (which play a central role for biinvolutive posets) are described.
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In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT−1, TBT−1). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set 0n,2,r,π ⊆ Mn × Mn (Mn = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø1,…,øs in the entries of A and B whose values are constant on all pairs similar in n,2,r,π to (A, B). The values of the functions øi(A, B), i = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in n,2,r,π. Let Sn be the subspace of complex symmetric matrices in Mn. For (A, B) ϵSn × Sn we consider the similarity class (TATt, TBTt), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ϵSn × Sn.
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This expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices. The treatment is in the spirit of the well-known book of Gantmacher on matrix theory, and may be regarded as a supplement to Gantmacher's chapters on pencils of matrices.
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We obtain a real canonical form for real symplectic pencils. It extends earlier results which were derived over the complex field, so that the canonical form was complex, and which were limited to the case where the elementary divisors of eigenvalues on the unit circle have an even degree.
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Matrices A and B are said to be unitarily similar if U∗AU = B for some unitary matrix U. This expository paper surveys results on canonical forms and invariants for unitary similarity. The first half gives a detailed description of methods developed by several authors (Brenner, Littlewood, Mitchell, McRae, Radjavi, Sergeĭchuk, and Benedetti and Cragnolini) using inductively defined reduction procedures to transform matrices to canonical form. The matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form. At each stage, one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks. The process ends in a finite number of steps, producing both the canonical form and the subgroup of the unitary group that preserves that form. Depending on the initial step, various canonical forms may be defined. The method can also be used to define canonical forms relative to certain subgroups of the unitary group, and canonical forms for finite sets of matrices under simultaneous unitary similarity. The remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity, such as the Specht-Pearcy trace invariants, the numerical range, and unitary reducibility.
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We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.
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Canonical forms for congruence and ∗congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347–353], based on Sergeichuk’s paper [Math. USSR, Izvestiya 31 (3) (1988) 481–501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous ∗congruence of pairs of complex Hermitian matrices.