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Multiplicative and additive compounds via Kronecker products and Kronecker sums

Ron Ofir, Michael Margaliot

Abstract

Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use. Here, we derive new formulas for the multiplicative and additive compounds of a matrix using Kronecker products and sums. This provides a new approach to matrix compounds based on the well-known and powerful theory of Kronecker products and sums. We demonstrate several applications of these new formulas, including deriving a new expression for the additive compound of the product of two matrices.

Multiplicative and additive compounds via Kronecker products and Kronecker sums

Abstract

Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use. Here, we derive new formulas for the multiplicative and additive compounds of a matrix using Kronecker products and sums. This provides a new approach to matrix compounds based on the well-known and powerful theory of Kronecker products and sums. We demonstrate several applications of these new formulas, including deriving a new expression for the additive compound of the product of two matrices.
Paper Structure (17 sections, 15 theorems, 128 equations, 2 figures)

This paper contains 17 sections, 15 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compound matrices
  4. $k$-compounds and $k$-parallelotopes
  5. Kronecker product
  6. The Kronecker sum
  7. Main results
  8. The matrices $M_{n,k}$ and $L_{n,k}$
  9. Properties of the matrices $M_{n,k}$ and $L_{n,k}$
  10. Algebraic expressions for the $k$-multiplicative compound
  11. Algebraic expression for the $k$-additive compound
  12. Applications
  13. A new proof for the expression for the entries of $A^{[k]}$
  14. Generalizing an identity for $2\times 2$ matrices to the $n\times n$ case
  15. A new formula for the $k$-additive compound of a product of matrices
  16. ...and 2 more sections

Key Result

Lemma 1

The matrices $M_{n,k}$ and $L_{n,k}$ satisfy

Figures (2)

  • Figure 1: A 3D parallelotope in $\mathbb R^3$.
  • Figure 2: Multiplication by the matrices matrices $M_{3,2}$ and $L_{3,2}$. Note in particular that $L_{3,2} M_{3,2} abc= abc.$

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Example 1
  • Example 2
  • Definition 3
  • Example 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Example 4
  • ...and 21 more