Eigenschemes and the Jordan canonical form

Abstract

We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint.Comment: 23 pages; v3: minor corrections, final version as appeared in Linear Algebra App

On tensors that are determined by their singular tuples

“…This opens a new perspective to tensor optimization, and a possible direction to obtain a general notion of the Eckart-Young Theorem. Indeed, this theme has been studied in many works, we suggest [1], [4], [5], [7], [8], [14], [15], [16], [19], [20] for a clearer picture of the topic.…”

On Rees algebras of 2‐determinantal ideals

“…□ Lemma 7.8. All S-pairs of bidegree (2,1) To summarize, the highest monomial in either side of (7.1) occurs exactly once, respectively, in LEN 𝜀,𝜙,𝜓 and in UEN 𝛼,𝛽,𝛾 . The second highest monomial also occurs once; in the right-hand side it occurs in UEN 𝛼,𝛽,𝛾 , while in the left-hand side it does not occur in LEN 𝜀,𝜙,𝜓 .…”

Best rank k approximation for binary forms

“…Suppose f ∈ Sym d V a symmetric tensor and dimV = 2. We denote by Z the scheme defined by the polynomial D(f ), embedded in P(Sym d V ) by the d-Veronese embedding in PV (see [1] for the case of matrices).…”