Basic classes of matrices with respect to quaternionic indefinite inner product spaces

Alpay, Daniel; Ran, André C. M.; Rodman, Leiba

doi:10.1016/j.laa.2005.11.010

Cited by 13 publications

(13 citation statements)

References 36 publications

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“…In Sections 4 and 5 we prove Theorems 2.1 and 2.2. In Section 6 we present Theorem 5.4 of [37] about the wildness of the problem of classifying pairs (1) in which B may be degenerate.…”

Section: Introductionmentioning

confidence: 99%

Canonical matrices of isometric operators on indefinite inner product spaces

Sergeichuk

2008

Linear Algebra and its Applications

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We give canonical matrices of a pair (A, B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax, Ay) = B(x, y) on a vector space over F in the following cases:• F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric;• F is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F.These classification problems are wild if B may be degenerate. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].

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“…In Sections 4 and 5 we prove Theorems 2.1 and 2.2. In Section 6 we present Theorem 5.4 of [37] about the wildness of the problem of classifying pairs (1) in which B may be degenerate.…”

Section: Introductionmentioning

confidence: 99%

Canonical matrices of isometric operators on indefinite inner product spaces

Sergeichuk

2008

Linear Algebra and its Applications

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“…To prove the maximality of (2.7), we first notice that where H j is a restriction of H to N j (i.e., H j is a matrix relative to a fixed basis in N j such that [x, y] Hj = [x, y] H for all x, y ∈ N j ). Observe that each H j is invertible, in view of invertibility of H and part (1). Equality (2.8) is easily verified using the (H, φ)-orthogonality property established in (1).…”

Section: Thenmentioning

confidence: 88%

“…Observe that each H j is invertible, in view of invertibility of H and part (1). Equality (2.8) is easily verified using the (H, φ)-orthogonality property established in (1). Thus,…”

Section: Thenmentioning

confidence: 88%

“…Part (1). The existence of such S, for some signs ζ 1 and ζ 2 follows from the canonical form (8.1), (8.2).…”

Section: 4mentioning

confidence: 93%

“…For the case (III) the result of Proposition 1.2 is well known, see [6] and [7], for example; see also [1], where a proof is given in the context of quaternion hermitian matrices. Case (V) is easily reduced to that context using the easily verifiable equality…”

Section: Is a Maximal (H φ)-Neutral Subspace If And Only If: (A) Dimmentioning

confidence: 99%

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Invariant neutral subspaces for Hamiltonian matrices

Rodman¹

2014

ELA

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Abstract. Hamiltonian matrices with respect to a nondegenerate skewsymmetric or skewhermitian indefinite inner product in finite dimensional real, complex, or quaternion vector spaces are studied. Subspaces that are simultaneously invariant for the matrices and neutral in the indefinite inner product are of special interest. The dimension of maximal (by inclusion) such subspaces is identified in terms of the canonical forms and sign characteristics. Criteria for uniqueness of maximal invariant neutral subspaces are given. The important special case of invariant Lagrangian subspaces is treated separately. Comparisons are made between real, complex, and quaternion contexts; for example, for complex Hamiltonian matrices with respect to a nondegenerate skewhermitian inner product in a finite dimensional complex vector space, the (complex) dimension of (complex) maximal invariant neutral subspaces is compared to the (quaternion) dimension of (quaternion) maximal invariant neutral subspaces, and necessary and sufficient conditions are given for the two dimensions to coincide (this is not always the case).

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