A note on spaces of symmetric matrices

Causin, Andrea; Pirola, Gian Pietro

doi:10.1016/j.laa.2007.05.038

Cited by 7 publications

(9 citation statements)

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“…5 Moreover, if V contains the scalars, then V is invertible if and only if V is polynomially closed [13]. Besides this, we do not have a good understanding of the structure of invertible matrix subspaces as a subset of Gr k (C n×n ).…”

Section: Theorem 33 ([13])mentioning

confidence: 97%

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Differential geometry of matrix inversion

Huhtanen¹

2010

MATH. SCAND.

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Essentially, there exists just the dimension segregating (square) matrix subspaces. In view of algebraic operations, this quantity is not particularly descriptive. For differential geometric information on matrix inversion, the second fundamental form is found for the set of inverses of the invertible elements of a matrix subspace. Several conditions for this form to vanish are given, such as being equivalent to a Jordan subalgebra. Global measures of curvature are introduced in terms of an analogy of the Nash fiber.

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Section: Theorem 33 ([13])mentioning

confidence: 97%

“…Otherwise, V is called singular if it does not contain invertible elements. (For singular matrix subspaces, see [5] and references therein.) For a nonsingular matrix subspace V , set Inv(V ) = {V −1 : V ∈ V ∩ GL(n, C)}, where GL(n, C) denotes the group of invertible n-by-n complex entried matrices.…”

Section: Introductionmentioning

confidence: 99%

Differential geometry of matrix inversion

Huhtanen¹

2010

MATH. SCAND.

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“…For the (restricted) instanton of charge 2 we can say a little more. The m-effective pair associated to E is in this case (4,6). In the recent work [1] it is shown that the only possibilities for such a pair are to be associated TOME 65 (2015), FASCICULE 5 to either a quotient of…”

Section: Remark 24 -If Cokermentioning

confidence: 99%

“…Notre résultat étend des travaux antérieurs sur le cas c 1 3. happened for example in [14,16,15,3]. On the other side, linear systems have been proved useful in approaching some classical problems in geometry: examples of this phenomenon are, among many others, [4,10,5]. The main connection between the two areas comes from interpreting such a linear system as a vector bundles map, whose kernel and cokernel are again vector bundles on some projective variety.…”

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confidence: 99%

Planes of matrices of constant rank and globally generated vector bundles

Boralevi¹,

Mezzetti²

2015

Ann. inst. Fourier

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We consider the problem of determining all pairs (c1, c2) of Chern classes of rank 2 bundles that are cokernel of a skew-symmetric matrix of linear forms in 3 variables, having constant rank 2c1 and size 2c1 + 2. We completely solve the problem in the "stable" range, i.e. for pairs with c 2 1 − 4c2 < 0, proving that the additional condition c2 ≤ c 1 +1 2 is necessary and sufficient. For c 2 1 − 4c2 ≥ 0, we prove that there exist globally generated bundles, some even defining an embedding of P 2 in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on c1 ≤ 3.2010 Mathematics Subject Classification. 14J60, 15A30. Key words and phrases. Skew-symmetric matrices, constant rank, globally generated vector bundles. Research partially supported by MIUR funds, PRIN 2010-2011 project "Geometria delle varietà algebriche", and by Università degli Studi di Trieste -FRA 2013 project "Geometria e topologia delle varietà".

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“…( 6 ) Otherwise a matrix subspace is called singular. (Such subspaces are also of interest [10], [8].) If there exists an invertible element in a subspace V of C n×n over C (or R), then the set of invertible elements is a dense and open subset of V [21, Theorem 2.2].…”

Section: Matrix Subspaces Possessing Invertible Elementsmentioning

confidence: 99%

Matrix subspaces and determinantal hypersurfaces

Huhtanen

2010

Ark. Mat.

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Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.

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A note on spaces of symmetric matrices

Abstract: We calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank generalizing a well-known result of Adams et al. AMS (MOS) Subject Classification: 15A30 (55N15).

Cited by 7 publications

References 20 publications

Differential geometry of matrix inversion

Differential geometry of matrix inversion

Planes of matrices of constant rank and globally generated vector bundles

Matrix subspaces and determinantal hypersurfaces

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