Tensors with eigenvectors in a given subspace

The first author with B. Sturmfels studied the variety of matrices with eigenvectors in a given linear subspace, called 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-ples 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.


Introduction
We introduce the subject of this paper with a basic example. Let L ⊂ C n be a linear space of dimension d. We are interested in the variety K d,n,m (L) of matrices A ∈ C n ⊗ C m having a singular pair (v, w) ∈ C n × C m with v ∈ L. Namely A ∈ K d,n,m (L) if and only if there exists v ∈ C n \ {0}, w ∈ C m \ {0}, λ 1 , λ 2 ∈ C such that Aw = λ 1 v, A t v = λ 2 w and v ∈ L (over R it is well known that the equality λ 1 = λ 2 can be assumed but already on C the situation is more subtle, see [2] and Example 3.10, note that we have not conjugated the transpose matrix A t ). The main Theorem of this paper applies in this case (see Theorem 3.4 and Remark 3.7) and it shows that the variety K d,n,m (L) is algebraic, irreducible, has codimension n − d and it has degree (see Remark 3.7) For n ≤ m the expression (1.1) simplifies to 2 n−d n d−1 (see Proposition 3.6), which does not depend on m, in other words it stabilizes for m ≫ 0. Moreover, for n ≤ m, we have that A ∈ K d,n,m (L) if and only if the symmetric matrix AA t has an eigenvector in L (see Lemma 3.8), so that the result is a consequence of [15,Prop. 1.2]. The paper [15], by the first author and B. Sturmfels, was indeed the main source for this paper. When n > m the condition that AA t has an eigenvector on L is necessary for A ∈ K d,n,m (L) but no more sufficient (see Example 3.9) and [15,Prop. 1.2] cannot be invoked anymore.
The formula (1.1) can be expressed as the coefficient of the monomial in the polynomial Note that both fractions are indeed homogeneous polynomials, respectively of degree n − 1 and m − 1. We list the degree values for small d, n, m in Table 1 in §3. The above formulation generalizes to tensors and it reminds the way we have computed it and also the formula to compute the number of singular t-ples of a tensors obtained by S. Friedland and the first author in [5]. This was interpreted as EDdegree of the Segre variety in [3]. The techniques we use comes from Algebraic Geometry and corresponds to a Chern class computation of a certain vector bundle. Singular pairs replace eigenvectors for non symmetric matrices in optimization questions. Generalizing from matrices to tensors, we have two analogous concepts. For symmetric tensors in Sym k C n , eigenvectors [11,17] are used in best rank one approximation problems, while for general tensors in C n 1 ⊗. . .⊗C n k k-ples of singular vectors [11] play the same role. Here are our main results.
The variety κ d,n 1 ,...,n k (L) = {T ∈ P(C n 1 ⊗. . .⊗C n k )|T has a singular k-tuple (v 1 , . . . , v k ) with v 1 ∈ L} is irreducible, it has codimension n 1 − d and degree given by the coefficient of the The structure of this paper is as follows. In §2 we study Kalman varieties for symmetric tensors. In Proposition 2.1 we prove irreducibility and compute the dimension, in Theorem 2.3 we compute the degree. In §3 we study Kalman varieties for general tensors, in Proposition 3.2 we prove irreducibility and compute the dimension, we compute the degree in Theorem 3.4 and we prove a stabilization result when n k ≫ 0, off the boundary format, see Corollary 3.5.
We thank Luca Sodomaco for helpful discussions. The first author is member of GNSAGA-INDAM. The second author thanks the Department of Mathematics and Computer Science of the University of Florence for support and hospitality. Both authors acknowledge support from the H2020-MSCA-ITN-2018 project POEMA.

Symmetric tensors with eigenvectors in a given subspace
Let L ⊂ C n be a linear subspace of dimension d. A symmetric tensor T ∈ Sym k (C n ) can be represented by a homogeneous polynomial f T ∈ C[x 1 , . . . , x n ] of degree k given by . . x i k . Eigenvectors of matrices was extended to symmetric tensors by Lim [11] and Qi [17].
and v are dependent. This is the definition used in [1] and [12], while in [18] E-eigenvectors are defined with the additional requirement that are not isotropic. The general tensor has no isotropic eigentensors [4,Lemma 4.2], so that for general tensors the two definitions coincide. We are interested in the scheme κ s d,n,k (L) of all symmetric tensors f ∈ Sym k C n that have an eigenvector in L.
Proposition 2.1. The variety κ s d,n,k (L) is irreducible, it has codimension n − d . Proof. We regard vectors v ∈ C n \ {0} as points in the projective space P(C n ), and we regard a polynomial of degree k as a point in P(Sym k (C n )). The product of these two projective spaces, X = P(Sym k (C n )) × P(C n ), has the two projections Fix the incidence variety W = {(f, z) ∈ X|z is an eigenvector of f} The projection of W to the second factor Hence it is a linear subspace of codimension n − 1 in P(Sym k (C n )). These properties imply that W is irreducible and has codimension n − 1 in P(Sym k (C n )) × P(C n ).
The projection of the incidence variety W to the first factor is surjective and p −1 (f ) = {z ∈ P(C n )|z is an eigenvector of f}. This set is finite for generic f and its number is equal to (k − 1) n − 1 k − 2 by [6] or [1] (this expression simplifies to n for k = 2). We note that κ s d,n,k (L) = p(W ∩ q −1 (P(L))) ⊆ P(Sym k (C n )).
The (d − 1)−dimensional subspace P(L) of P(C n ) = P n−1 specifies the following diagram: Each fiber of the map q in above diagram is a linear space of codimension n − 1 in P(Sym k (C n )). This implies that W ∩ q −1 (P(L)) is irreducible and its dimension equals Since the general fiber of the surjection p is finite, the variety κ d,n,k (L) is irreducible of the same dimension. Hence κ s d,n,k (L) has codimension n − d in P ( k+n−1 k )−1 .
In order to compute the degree of κ s d,n,k (L) we construct a vector bundle E on X with a section vanishing on W . We briefly recall the construction in [12, §3.1].
Let O(−1) be the universal bundle of rank 1 and Q the quotient bundle of rank n − 1. They appear in the following exact sequence, with O the structure sheaf of P n−1 : The section s f vanishes on v if and only if π(f (v k−1 )) = 0. In particular, every symmetric tensor f ∈ Sym k C n defines (by contraction) an element in Hom(Sym k−1 C n , C n ) and hence a section of Q(k − 1) which by abuse of notation we denote again with s f . This implies We recall that if a vector bundle E of rank r on X has a section vanishing on Z, and the codimension of Z is equal to r, then the class of [Z] in the degree r component of the Chow shall apply this to the following vector bundle on the product variety X Since H 0 (O(1)) = Sym k (C n ) and H 0 (Q(k − 1)) = Γ k,1 n−2 C n , by Künneth formula we get H 0 (E) = Hom(Sym k (C n ), Γ k,1 n−2 C n ).
We have a section I ∈ H 0 (E) given by the map M → s M . The section I vanishes exactly at the pairs (M, z) such that z is eigenvector of M, so we get that the zero locus Z(I) of I ∈ H 0 (E) equals the incidence variety W .
Proof. Since Z(I) has codimension rkE = n − 1 in X, the class of Z(I) equals the top chern class of E(By theorem 4.4). In symbols, [Z(I)] = c n−1 (E). The desired degree equals ). Hence the equality on the right of (2.4) can be written as All summands are zero except for i = d − 1, and we remain with deg c d−1 (Q(k − 1)).
Since deg c i (Q) = 1, rk(Q) = n − 1, the result follows from the formula Remark 2.4. The result generalizes to any complex vector space V equipped with a symmetric nondegenerate bilinear form. The construction works in the setting of SO(V )-actions, in particular the space of sections we have considered, like H 0 (Q(k− 1)), are SO(V )-modules. Note that in this setting V is isomorphic to its dual V ∨ .

Singular vector k-ples of tensors
Let L ⊂ C n 1 be a fixed d−dimensional linear subspace .
We are interested in the Kalman variety κ d,n 1 ,...,n k (L) of all tensors T ∈ C n 1 ⊗ . . . ⊗ C n k that have a singular k−tuple (v 1 , . . . , v k ) with v 1 ∈ L.
Proof. The product X = P(C n 1 ⊗ . . . ⊗ C n k ) × P(C n 1 ) × . . . × P(C n k ), has the two projections Fix the variety (we use the same notation of previous section to ease the analogy) The projection of W to the second factor is surjective and every fiber is and this is the zero scheme of the ideal of 2 × 2 minors of the n i × 2 matrix The projection of the variety W to the first factor The Kalman variety has the following description in terms of the above diagram: κ d,n 1 ,...,n k (L) = p(W ∩ q −1 (q −1 1 (P(L)))) where q −1 1 (P(L)) = P(L) × P(C n 2 ) × . . . × P(C n k ). The (d − 1)−dimensional subspace P(L) of P(C n 1 ) = P n 1 −1 specifies the following diagram: Each fiber of the map q in above diagram is a linear space of codimension Σ k i=1 (n i −1) in P((C n 1 ⊗ . . . ⊗ C n k ), and we conclude exactly as in the proof of Proposition 2.1.
In order to compute the degree of κ d,n 1 ,...,n k (L) we construct a vector bundle E on X with a section vanishing exactly on W . We briefly recall the construction in [5, §3].
This implies Consider the following vector bundle on the product variety Hence We have a section I ∈ H 0 (E) given by the diagonal map T → (s T , . . . , s T ). The section I vanishes exactly at (T, v 1 , . . . , v k ) such that (v 1 , . . . , v k ) is a singular k-ple of T , so we get that the zero locus Z(I) of I ∈ H 0 (E) equals the incidence variety W .
Theorem 3.4. The degree of Kalman variety κ d,n 1 ,...,n k (L) is the coefficient of Proof. As in the proof of Theorem 2.3, the degree is equal to E is the direct sum of k summands, by the Euler sequence each of them has Chern polynomial (1+v 1 +...+ v i +...v k +h) n i (1+v 1 +...−v i +...v k +h) for i = 1, . . . , k. So the degree is the coefficient of Hence it is the coefficient of We get that the degree sought is the coefficient of All terms of the last polynomial have degree ≤ i (n i − 1) which equals the degree of , hence it is enough to consider the homogeneous part of top degree. The thesis follows.
The following stabilization phenomenon is similar to the one observed in [16].
Corollary 3.5. (Stabilization) Let (n k − 1) = k−1 i=1 (n i − 1) (boundary format, see [7]). For any m ≥ n k we have κ d,n 1 ,...,n k (L) = κ d,n 1 ,...,n k−1 ,m (L) The first one is the coefficient of With j < m − n k in the last sum we have Hence the two coefficients are equal.
This concludes the proof of the Corollary.
The following Proposition covers the case of matrices, k = 2.
Remark 3.7. Following Theorem 3.4, the coefficient of h n−d v d−1 w m−1 in the expansion of the polynomial is computed as (1.1) by Newton expansion. Lemma 3.8. Let A be a n × m matrix with n ≤ m. A has a singular pair (v, w) with v ∈ L if and only if AA t has an eigenvector in L.
In case A t v = 0, since n ≤ m we have ker A = 0 then with both λ i = 0 any w ∈ ker A works.
Example 3.9. The assumption n ≤ m is necessary in Lemma 3.8. Let A = 1 0 so AA t has a eigenvector in L but Aw ∈ L only if w = 0.
Example 3.10. If A has a singular pair (v, w) with v ∈ L then v is eigenvector of AA t , so AA t has an eigenvector in L. The converse is true on real numbers but it is false on complex numbers if we wish the additional requirement λ 1 = λ 2 . A simple counterexample is as follows.
A has no singular pair (v, w) with v ∈ L, but AA t = 0 so AA t has an eigenvector in L. Note the condition on rk(CA) is empty when n ≤ m.
Proof. The pair v 0 ⊗ (A t v 0 ) is a singular pair if (CA)(A t v 0 ) = 0, in order to be a nonzero pair we have to exclude the case when n > m and CA has maximal rank m.
We list in Table 1 the first cases of degree of κ d,n,m (L) for n ≥ m and its singularities. The singular locus of κ d,n,m (L) contains the closure of the set of matrices having at least two distinct singular pairs with first component on L. The table shows that this containment is strict in many cases, contrary to the eigenvector case in [15,Theorem 4.4], where equality holds.
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.