Matrix tuples with linearly dependent invariant subspaces

Abstract: The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes previous joint work.

• The paper presents the explicit construction of an irreducible hypersurface defined by the vanishing of a determinantal polynomial that characterizes matrix tuples with dependent invariant subspaces.
• It generalizes previous pairwise results by treating l-tuples of n×n matrices and employing spectral analysis and Grassmannian techniques to derive necessary and sufficient conditions.
• The research offers a computable algebraic certificate with precise degree calculations, providing insights for applications in systems theory, control, and quantum systems.

Matrix Tuples with Linearly Dependent Invariant Subspaces

Overview and Motivation

This paper addresses the algebraic and geometric characterization of tuples of square complex matrices that admit invariant subspaces whose dimensions sum to the ambient space dimension but do not collectively span the entire space. The analysis generalizes previous results regarding matrix pairs with nontrivially intersecting invariant subspaces to the setting of $l$-tuples of $n \times n$ complex matrices, for general partitions of $n$. The central achievement is the explicit construction of the irreducible hypersurface in the space of matrix tuples defined by such linear dependencies, and the derivation of its defining polynomial equation.

Definitions, Techniques, and Main Results

The setting involves tuples $(A_1,\ldots,A_l)$ of $n \times n$ complex matrices, and for each $i$, a prescribed dimension $k_i$ such that $\sum_{i=1}^l k_i = n$. The locus of interest is those tuples where one can find $A_i$-invariant subspaces $V_i$ of dimension $n \times n$0 (for each $n \times n$1) such that $n \times n$2.

The core construct is a polynomial $n \times n$3, defined as the determinant of a matrix $n \times n$4 built from a canonical alternating $n \times n$5-form $n \times n$6 (or its induced image under the canonical wedge-product map) and powers of a combined operator $n \times n$7 acting on the tensor product or wedge power of standard modules. The explicit structure aligns with classical themes in invariant theory and the representation theory of the general linear group.

A key observation is:

• Criterion: If, for a tuple $n \times n$8, there are $n \times n$9-invariant subspaces $n$0 as above whose span is deficient, then $n$1.
• Partial Converse: When the combined operator $n$2 has simple spectrum, the vanishing of $n$3 is equivalent to the existence of such invariant subspaces.

Spectral analysis reveals that $n$4 factors into irreducible invariant polynomials, denoted $n$5, indexed by all nonzero tuples $n$6 with $n$7, $n$8, describing partitions of the eigenvalues among the matrices. The factorization is crucial for pinpointing the hypersurface associated precisely with the desired geometric condition.

The main theorem demonstrates that the locus $n$9 of matrix tuples with this property is the vanishing set of the irreducible polynomial

$(A_1,\ldots,A_l)$0

and this polynomial is explicitly constructed and shown to be irreducible, confirming that $(A_1,\ldots,A_l)$1 is an irreducible projective variety.

Algebraic Geometry and Degree Calculations

The author employs advanced techniques from algebraic geometry, including arguments around Grassmannians $(A_1,\ldots,A_l)$2, projective varieties, and the behavior of multihomogeneous and bihomogeneous determinant polynomials. Irreducibility is established via dense orbit arguments for group actions and explicit determinantal representations.

A notable technical achievement is the precise computation of the degree of $(A_1,\ldots,A_l)$3:

$(A_1,\ldots,A_l)$4

matching the combinatorics of choosing invariant subspaces and measuring the codimension of their spans.

Numerical Strengths and Explicit Claims

• The hypersurface of interest is characterized by a unique irreducible polynomial whose degree is explicitly determined.
• The construction yields a computable determinantal obstruction for any specified tuple $(A_1,\ldots,A_l)$5.
• The analysis encompasses the full generality of partitions, not just the extremal or pairwise case.
• The polynomial criteria are shown to be both necessary and sufficient (given simple spectrum), extending previously known results for pairs to arbitrary tuples.

Theoretical and Practical Implications

The paper makes a substantial contribution to the invariant theory of matrix tuples and the intersection theory of invariant subspaces. The main result provides an effective tool for diagnosing linear dependencies among invariant subspaces across several matrices—a property of interest in systems theory, control, and theoretical physics, wherever symmetry breaking and simultaneous block— or triangularizability are studied.

From a theoretical perspective, the explicit description of the hypersurface enriches the understanding of the algebraic structure underpinning problems in simultaneous similarity, representation stability, and eigenstructure rigidity. Furthermore, the precise factorization of $(A_1,\ldots,A_l)$6 clarifies which components of the vanishing locus correspond to non-generic degeneracies (e.g., multiple eigenvalues), and which to the fundamentally geometric configurations of subspaces.

Practically, the results enable algebraic certificates for the non-generic intersection of invariant subspaces—these may be leveraged in computational settings for algorithms seeking block decompositions, or, for instance, when analyzing coupled dynamics or quantum systems.

Future Directions

Potential avenues for further research include:

• Extension of the approach to fields of positive characteristic, where invariant theory may exhibit qualitatively different phenomena.
• Investigation of related loci for other collections of subspaces, such as non-invariant or partially invariant flags.
• Study of the singularities of the hypersurface defined by $(A_1,\ldots,A_l)$7, and their geometric or representation-theoretic significance.
• Development of computational methods to evaluate the defining polynomial in practical scenarios involving large $(A_1,\ldots,A_l)$8, leveraging the determinantal construction.

Conclusion

This work systematically generalizes the algebraic characterization of matrix tuples with linearly dependent invariant subspaces, providing explicit, irreducible defining equations for the associated hypersurfaces in the space of tuples of complex matrices. The rigorous factorization of the relevant polynomial obstructions, precise degree computation, and robust algebraic-geometric arguments establish a strongly grounded framework for further inquiry into the intersection theory of invariant subspaces and its numerous applications.