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Rank of the derivative of the projection to symmetrized polydisc

Tran Duc Anh

Abstract

We prove that the rank of the derivative of the projection from spectral unit ball to symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take derivative.

Rank of the derivative of the projection to symmetrized polydisc

Abstract

We prove that the rank of the derivative of the projection from spectral unit ball to symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take derivative.

Paper Structure

This paper contains 9 sections, 1 theorem, 34 equations.

Table of Contents

  1. Introduction
  2. Content of the research
  3. Notations and local lifting problem
  4. Main result
  5. Jordan normal form and notations
  6. Proof of the main result
  7. The rank is less than or equal to the degree
  8. The rank is bigger than or equal to the degree
  9. Conclusion

Key Result

Theorem 2.2.1

Consider the symmetrization mapping $\pi\colon \Omega_n\to \mathbb{G}_n.$ Then for any $B\in \Omega_n,$ we have the rank of the derivative of $\pi$ at $B$ is equal to the degree of the minimal polynomial of $B.$

Theorems & Definitions (4)

  • Definition 2.1.1
  • Definition 2.1.2: local lifting problem
  • Theorem 2.2.1
  • Definition 2.2.2