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Nontrivial invariant subspaces of linear operator pencils
Part of
- Special classes of linear operators
- General theory of linear operators
Jaewoong Kim, Jasang Yoon
Journal:

Canadian Mathematical Bulletin / Volume 67 / Issue 1 / March 2024

Published online by Cambridge University Press:

07 June 2023, pp. 1-8

Print publication:

March 2024
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In this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair $\mathbf {T}=(T_{1},T_{2})$ and investigate nontrivial invariant subspaces between the generalized spherical Aluthge transform of the linear pencil of $\mathbf {T}$ and the linear pencil of the original pair $\mathbf {T}$ of bounded operators with dense ranges.

2 - Algebraic Eigenproblems and Their Applications
from Part I - Matrix Methods
Kevin W. Cassel, Illinois Institute of Technology
Book:

Matrix, Numerical, and Optimization Methods in Science and Engineering

Published online:

18 February 2021

Print publication:

04 March 2021, pp 64-154
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Summary

The algebraic eigenproblem is the mathematical answer to the physical questions:What are the principal stresses in a solid or fluid and on what planes do they act?What are the natural frequencies of a system?Is the system stable to small disturbances?What is the best basis with respect to which to solve a system of linear algebraic equations with a real symmetric coefficient matrix?What is the best basis with respect to which to solve a system of linear ordinary differential equations?What is the best basis with respect to which to represent an experimental or numerical data set?

POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn
Part of
- Classical measure theory
- Geometric probability and stochastic geometry
S. REZA MOGHADASI
Journal:

Bulletin of the Australian Mathematical Society / Volume 85 / Issue 2 / April 2012

Published online by Cambridge University Press:

06 January 2012, pp. 315-324

Print publication:

April 2012
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The Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.