Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real-congruence canonical forms of real matrices

Fernando De Terán, Froilán M. Dopico

TL;DR

This work introduces two new canonical forms for real matrices under real-congruence: a four-block-type form derived from the Horn–Sergeichuk $^*$-congruence canonical form and a three-block-type, block-tridiagonal form derived from the Futorny–Horn–Sergeichuk $^*$-congruence canonical form. Each form provides a structured, unique representative for the real-congruence class and is explicitly connected to the real-Kronecker canonical form of the pair $(A^ op,A)$. The authors establish precise relations between the two real-congruence forms, relate them to the Lee–Weinberg eight-type form, and supply explicit correspondences between blocks via real-KCF mappings. The results yield practical tools for classifying bilinear forms over $\mathbb{R}$ and elucidate how real-congruence structures encode the same information as the associated real palindromic pencils.

Abstract

We present two new canonical forms for real congruence of a real square matrix $A$. The first one is a direct sum of canonical matrices of four different types and is obtained from the canonical form under $^*$congruence of complex matrices provided by Horn and Sergeichuk in [Linear Algebra Appl. 416 (2006) 1010-1032]. The second one is a direct sum of canonical matrices of three different types, has a block tridiagonal structure and is obtained from the canonical form under $^*$congruence of complex matrices provided by Futorny, Horn and Sergeichuk in [J. Algebra 319 (2008) 2351-2371]. A detailed comparison between both canonical forms is also presented, as well as their relation with the real Kronecker canonical form under strict real equivalence of the matrix pair $(A^\top , A)$. Another canonical form for real congruence was presented by Lee and Weinberg in [Linear Algebra Appl. 249 (1996) 207-215], which consists of a direct sum of eight different types of matrices. In the last part of the paper, we explain the correspondence between the blocks in this canonical form and those in the two new forms introduced in this work.

Real-congruence canonical forms of real matrices

TL;DR

This work introduces two new canonical forms for real matrices under real-congruence: a four-block-type form derived from the Horn–Sergeichuk -congruence canonical form and a three-block-type, block-tridiagonal form derived from the Futorny–Horn–Sergeichuk -congruence canonical form. Each form provides a structured, unique representative for the real-congruence class and is explicitly connected to the real-Kronecker canonical form of the pair . The authors establish precise relations between the two real-congruence forms, relate them to the Lee–Weinberg eight-type form, and supply explicit correspondences between blocks via real-KCF mappings. The results yield practical tools for classifying bilinear forms over and elucidate how real-congruence structures encode the same information as the associated real palindromic pencils.

Abstract

We present two new canonical forms for real congruence of a real square matrix . The first one is a direct sum of canonical matrices of four different types and is obtained from the canonical form under congruence of complex matrices provided by Horn and Sergeichuk in [Linear Algebra Appl. 416 (2006) 1010-1032]. The second one is a direct sum of canonical matrices of three different types, has a block tridiagonal structure and is obtained from the canonical form under congruence of complex matrices provided by Futorny, Horn and Sergeichuk in [J. Algebra 319 (2008) 2351-2371]. A detailed comparison between both canonical forms is also presented, as well as their relation with the real Kronecker canonical form under strict real equivalence of the matrix pair . Another canonical form for real congruence was presented by Lee and Weinberg in [Linear Algebra Appl. 249 (1996) 207-215], which consists of a direct sum of eight different types of matrices. In the last part of the paper, we explain the correspondence between the blocks in this canonical form and those in the two new forms introduced in this work.

Paper Structure

This paper contains 17 sections, 30 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation, basic notions, and basic results
  3. Congruence of matrices. Equivalence of matrix pairs
  4. Canonical blocks
  5. Kronecker and real-Kronecker canonical forms of matrix pairs
  6. Horn-Sergeichuk canonical form for $^*$congruence
  7. Futorny-Horn-Sergeichuk tridiagonal canonical form for $^*$congruence
  8. Relation between Horn-Sergeichuk and Futorny-Horn- Sergeichuk canonical forms
  9. First canonical form of real matrices for real-congruence
  10. Relation of the first canonical form for real-congruence of $A$ and the real-KCF of $(A^\top,A)$
  11. Second canonical form of real matrices for real-congruence: block tridiagonal form
  12. Relation of the second canonical form for real-congruence of $A$ and the real-KCF of $(A^\top, A)$
  13. Relation between the first and the second canonical forms for real congruence
  14. Correspondence with the blocks in the Lee-Weinberg canonical form
  15. Proof of Theorem \ref{['thm.correspondlw96']}
  16. ...and 2 more sections

Key Result

Theorem 2.2

di02 Let $A,B\in\mathbb R^{n\times n}$ be such that $PAP^* = B$, for some invertible $P\in\mathbb C^{n\times n}$. Then, there exists an invertible $Q\in\mathbb R^{n\times n}$ such that $QAQ^\top=B$.

Theorems & Definitions (52)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 42 more