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Invariant subspace perturbation of a matrix with Jordan blocks

Hongguo Xu

Abstract

We investigate how invariant subspaces corresponding to a single eigenvalue will change when a matrix is perturbed. We focus on the invariant subspaces corresponding to an eigenvalue associated with the Jordan blocks that have the same size. An invariant subspace can be expressed as the range of a full column matrix. We characterize the perturbed invariant subspaces with such matrices expressed in a sum form that exhibits the fractional orders. We also provide the formulas for the coefficient matrices associated with the zero and first fractional orders. The results extend the standard invariant subspace perturbation theory.

Invariant subspace perturbation of a matrix with Jordan blocks

Abstract

We investigate how invariant subspaces corresponding to a single eigenvalue will change when a matrix is perturbed. We focus on the invariant subspaces corresponding to an eigenvalue associated with the Jordan blocks that have the same size. An invariant subspace can be expressed as the range of a full column matrix. We characterize the perturbed invariant subspaces with such matrices expressed in a sum form that exhibits the fractional orders. We also provide the formulas for the coefficient matrices associated with the zero and first fractional orders. The results extend the standard invariant subspace perturbation theory.
Paper Structure (5 sections, 7 theorems, 167 equations)

This paper contains 5 sections, 7 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant subspace perturbation
  4. First fractional order invariant subspace perturbation
  5. Conclusions

Key Result

Theorem 1

Suppose $A$ and $D$ are transformed to (ta) and (td), where $A_{11}$ and $D_{11}$ are given in (nila) and (blkb), respectively, and $t>0$ is a sufficiently small number.

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Example 1