[Paper Review] Matrix Representations for Symmetric and Antisymmetric Multi-Linear Maps

Ural Bekbaev|arXiv (Cornell University)|Oct 13, 2010

Advanced Topics in Algebra2 references3 citations

TL;DR

This paper presents a traditional matrix representation for symmetric and antisymmetric multi-linear maps using column vectors, reformulating earlier work that used row vectors. It introduces a novel matrix product operation that preserves algebraic structure and establishes normed spaces for convergence analysis, enabling systematic computation and theoretical analysis of multi-linear maps in linear algebra and polynomial mappings.

ABSTRACT

In this paper the main results in arXiv:0901.3179v3, related to the matrix representation of polynomial maps, are restated in traditional way of linear algebra assuming that variable vectors are presented as column vectors. Some new results related to that subject are also included. Here one can find the behavior of the matrices of polynomial maps with respect to the change of variables (coordinate system), matrix representations for symmetric and antisymmetric multi-linear maps. It is shown also that the offered representations are in good concordance with known operations over such multi-linear maps.

Motivation & Objective

• To reformulate matrix representations of polynomial and multi-linear maps using column vectors, aligning with standard linear algebra conventions.
• To introduce a new matrix product operation that preserves algebraic properties such as associativity and distributivity.
• To provide matrix representations for symmetric and antisymmetric multi-linear maps in a way compatible with known operations.
• To establish a normed structure on matrix spaces to analyze convergence of multi-variable power series.
• To demonstrate that the proposed matrix framework is consistent with wedge products and exterior algebra operations.

Proposed method

• The paper defines multi-indices and uses a lexicographic ordering on non-negative integer tuples to organize matrix entries.
• It introduces a specialized matrix product $ A \bigodot B $ that combines entries via multinomial coefficients and supports composition of multi-linear maps.
• The matrix product is shown to satisfy key algebraic properties: commutativity, associativity, distributivity, and compatibility with scalar multiplication.
• The paper defines a norm $ \|A\|_{\rho} $ on matrix spaces to analyze convergence, using Hölder's inequality to prove submultiplicativity.
• It establishes a correspondence between multi-linear maps and matrices via symmetric and antisymmetric extensions using wedge products.
• The framework is validated by showing that matrix representations of wedge products match known algebraic identities.

Experimental results

Research questions

• RQ1 How can symmetric and antisymmetric multi-linear maps be consistently represented using matrices when variable vectors are treated as column vectors?
• RQ2 What algebraic properties must a matrix product satisfy to preserve the structure of multi-linear maps under composition?
• RQ3 Can a norm be defined on matrix spaces of multi-linear maps such that the norm of a composed map is bounded by the product of norms?
• RQ4 How does the proposed matrix product relate to standard operations in exterior algebra, such as the wedge product?
• RQ5 To what extent does the new matrix representation preserve the behavior of polynomial maps under coordinate transformations?

Key findings

• The proposed matrix product $ A \bigodot B $ is associative, commutative, and distributive, and preserves the structure of multi-linear maps under composition.
• The norm $ \|A\|_{\rho} $ satisfies the triangle inequality and is submultiplicative: $ \|A \wedge B\| \leq \|A\| \|B\| $, ensuring convergence control for power series.
• For any matrix $ A $, the $ m $-th power under the new product satisfies $ (h^{(m)})_{0,\alpha'} = m! h^{\alpha'} $, linking matrix powers to monomial coefficients.
• The matrix of the wedge product $ \textbf{A} \wedge_{\textbf{C}} \textbf{B} $ is given by $ C(\overline{A} \wedge \underline{B}) $, confirming consistency with exterior algebra.
• The matrix representation of a symmetric multi-linear map is invariant under permutation of indices, and the representation is compatible with symmetric tensor products.
• The framework allows for the computation of the radius of absolute convergence of multi-variable power series via matrix norm analysis.