[Paper Review] Pasting and Reversing Operations over some Vector Spaces

Primitivo B. Acosta-Humánez, Adriana L. Chuquen+1 more|arXiv (Cornell University)|Sep 19, 2012

Scheduling and Optimization Algorithms3 references3 citations

TL;DR

This paper introduces and formalizes the operations of Pasting and Reversing over vector spaces, including vectors, matrices, and polynomials, establishing that the sets of palindromic and antipalindromic vectors form complementary subspaces. The key contribution is proving that any vector space decomposes as a direct sum of its palindromic and antipalindromic subspaces, with dim(V) = dim(Wₚ) + dim(Wₐ).

ABSTRACT

Pasting and Reversing operations have been used successfully over the set of integer numbers, simple permutations, rings and recently over a generalized vector product. In this paper, these operations are defined from a natural way to be applied over vector spaces. In particular we study Pasting and Reversing over vectors, matrices and we rewrite some properties for polynomials as vector space. Finally we study some properties of Reversing through linear transformations as for example an analysis of palindromic and antipalindromic vector subspaces.

Motivation & Objective

• To formalize Pasting and Reversing operations over vector spaces, extending their use from integers and permutations to vectors, matrices, and polynomials.
• To investigate the algebraic structure of palindromic and antipalindromic vectors under linear operations.
• To establish that the set of palindromic vectors (Wₚ) and antipalindromic vectors (Wₐ) are vector subspaces of a given vector space V.
• To prove that V = Wₐ ⊕ Wₚ, implying dim(V) = dim(Wₐ) + dim(Wₚ), providing a structural decomposition of vector spaces.
• To explore applications of these operations in linear algebra education and potential extensions to mathematical physics and dynamical systems.

Proposed method

• Define Reversing of a vector v ∈ Kⁿ as ṽ = (vₙ, vₙ₋₁, ..., v₁), establishing it as an involution and linear operation.
• Prove that Reversing preserves dot products and vector cross products under reversal, i.e., v·w = ṽ·w̃ and ṽ×w̃ = w̃×ṽ for v,w ∈ K³.
• Introduce Pasting by blocks (⋄₆) for matrices A ∈ Mₙₓₘ(K) and B ∈ Mᵣₓₛ(K), defined as a block-diagonal matrix with A and B on the diagonal and zero blocks elsewhere.
• Establish properties of Reversing on matrices, including R(AB) = R(B)R(A), R(Aᵀ) = R(A)ᵀ, and det(R(A)) = det(A).
• Use block matrix theory to prove that R(A ⋄₆ B) = R(B) ⋄₆ R(A), and that (A ⋄₆ B)ᵀ = Aᵀ ⋄₆ Bᵀ.
• Apply the operations to polynomials by treating them as vectors in Kⁿ, thereby inheriting the same reversal and pasting structures.

Experimental results

Research questions

• RQ1 How can Pasting and Reversing be generalized from integers and permutations to vector spaces, particularly for vectors and matrices?
• RQ2 What algebraic structure do the sets of palindromic and antipalindromic vectors form within a vector space?
• RQ3 Under what conditions does the direct sum decomposition V = Wₐ ⊕ Wₚ hold, and what is its dimensionality implication?
• RQ4 How do Reversing and Pasting operations behave under matrix operations such as transpose, inverse, and determinant?
• RQ5 What are the implications of these operations for linear algebra education and applications in mathematical physics?

Key findings

• The set of palindromic vectors Wₚ and antipalindromic vectors Wₐ are both vector subspaces of V.
• The vector space V decomposes as a direct sum: V = Wₐ ⊕ Wₚ, implying that every vector in V can be uniquely written as the sum of a palindromic and an antipalindromic vector.
• The dimension of V equals the sum of the dimensions of its palindromic and antipalindromic subspaces: dim(V) = dim(Wₐ) + dim(Wₚ).
• Reversing is an involutive linear transformation: R(R(v)) = v and R(av + bw) = aR(v) + bR(w) for all a,b ∈ K and v,w ∈ V.
• For matrices, Reversing preserves the trace and determinant, and satisfies R(AB) = R(B)R(A), with R(Aᵀ) = R(A)ᵀ.
• Pasting by blocks (⋄₆) preserves matrix operations such as transpose, inverse, and determinant, with det(A ⋄₆ B) = det(A)det(B) and Tr(A ⋄₆ B) = Tr(A) + Tr(B).