# Pencils of Quadrics: Old and New

@article{Fevola2020PencilsOQ, title={Pencils of Quadrics: Old and New}, author={Claudia Fevola and Yelena Mandelshtam and B. Sturmfels}, journal={arXiv: Algebraic Geometry}, year={2020} }

Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We compute the reciprocal curve and the maximum likelihood degrees, and we study strata of pencils in the Grassmannian.

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#### References

SHOWING 1-10 OF 17 REFERENCES

Pencils of complex and real symmetric and skew matrices

- Mathematics
- 1991

Abstract 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… Expand

Skew-symmetric matrix pencils : codimension counts and the solution of a pair of matrix equations

- Mathematics
- 2013

The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution ...

Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations

- Mathematics
- 2014

The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In additio… Expand

A recurring theorem about pairs of quadratic forms and extensions: a survey

- Mathematics
- 1979

Abstract This is a historical and mathematical survey of work on necessary and sufficient conditions for a pair of quadratic forms to admit a positive definite linear combination and various… Expand

Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry

- Mathematics
- 2009

We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing… Expand

Linear systems of real quadratic forms. II

- Mathematics
- 1964

1. It is natural to consider invariantive properties of k-dimensional linear systems of real quadratic forms over Rn (or quadratic forms over Rn with values in RkI 2< k5 In(n + 1)), because of the… Expand

Classical Algebraic Geometry: A Modern View

- Mathematics
- 2012

Algebraic geometry has benefited enormously from the powerful general machinery developed in the latter half of the twentieth century. The cost has been that much of the research of previous… Expand

The dimension of matrices (matrix pencils) with given Jordan (Kronecker) canonical forms

- Mathematics
- 1995

Abstract The set of n by n matrices with a given Jordan canonical form defines a subset of matrices in complex n 2 dimensional space. We analyze one classical approach and one new approach to count… Expand

ON DOMINANCE AND VARIETIES OF COMMUTING MATRICES

- Mathematics
- 1961

[4]), a question first raised by Goto. In the same section it is shown that the algebra generated by a pair of commuting n x n matrices can not have dimension greater than n, and that this algebra… Expand

Methods of algebraic geometry

- Mathematics
- 1947

“THIS volume is the first part of a work designed it provide a convenient account of the foundations and methods of modem algebraic geometry.” These words from the authors' preface explain the scope… Expand