Equations Of Kalman Varieties

Huang, Hang

doi:10.48550/arxiv.1707.08699

Cited by 2 publications

(2 citation statements)

References 2 publications

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“…Remark 3.12. In the matrix case d = 2, equations for κ s d,n,m (L) were found in [20,10], even in the more general case of matrix eigenvectors. It should be interesting to extend that study to the tensor case.…”

Section: The Fiber Of Qmentioning

confidence: 99%

Tensors with eigenvectors in a given subspace

Ottaviani

Shahidi

2021

Rend. Circ. Mat. Palermo, II. Ser

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The first author with B. Sturmfels studied in [16] the variety of matrices with eigenvectors in a given linear subspace, called the Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore, we consider the Kalman variety of tensors having singular t-tuples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

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Section: The Fiber Of Qmentioning

confidence: 99%

Tensors with eigenvectors in a given subspace

Ottaviani

Shahidi

2021

Rend. Circ. Mat. Palermo, II. Ser

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“…They determined its codimension, degree and studied its singular locus. Thereafter, Sam [Sam12] and Huang [Hua17] determined their defining equations. More recently, Ottaviani and the first author [OS21] rephrased the original setting for singular vector pairs, extending it to the case of singular vector k-tuples.…”

Section: Introductionmentioning

confidence: 99%

Degrees of Kalman varieties of tensors

Shahidi,

Sodomaco,

Ventura

2021

Preprint

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Kalman varieties of tensors are algebraic varieties consisting of tensors whose singular vector k-tuples lay on prescribed subvarieties. They were first studied by Ottaviani and Sturmfels in the context of matrices. We extend recent results of Ottaviani and the first author to the partially symmetric setting. We describe a generating function whose coefficients are the degrees of these varieties and we analyze its asymptotics, providing analytic results à la Zeilberger and Pantone. We emphasize the special role of isotropic vectors in the spectral theory of tensors and describe the totally isotropic Kalman variety as a dual variety.

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Equations Of Kalman Varieties

Abstract: The Kalman variety of a linear subspace is a vector space consisting of all endomorphisms that have an eigenvector in that subspace. We resolve a conjecture of Ottaviani and Sturmfels and give the minimal defining equations of the Kalman variety over a field of characteristic 0.

Cited by 2 publications

References 2 publications

Tensors with eigenvectors in a given subspace

Tensors with eigenvectors in a given subspace

Degrees of Kalman varieties of tensors

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