On optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix

Area of Science:

• Linear Algebra
• Matrix Theory
• Numerical Analysis

Background:

• Skew circulant matrices are a specialized class of matrices with applications in various fields.
• Efficient methods for analyzing their properties, such as spectral decomposition and singular values, are crucial.
• Understanding matrix perturbations is essential for assessing the stability and accuracy of solutions to linear systems.

Purpose of the Study:

• To develop a style spectral decomposition for a special skew circulant matrix.
• To extend this decomposition to arbitrary skew circulant matrices using Kronecker products.
• To derive the singular values of skew circulant matrices and perform optimal backward perturbation analysis for associated linear systems.

Main Methods:

• Style spectral decomposition of a special skew circulant matrix.
• Utilizing Kronecker products to generalize the decomposition for arbitrary skew circulant matrices.
• Derivation of singular values and backward perturbation analysis.

Main Results:

• A novel style spectral decomposition for skew circulant matrices is established.
• The singular values of skew circulant matrices are obtained.
• Optimal backward perturbation analysis for linear systems with skew circulant coefficient matrices is performed.

Conclusions:

• The proposed spectral decomposition provides an effective tool for analyzing skew circulant matrices.
• The derived singular values and perturbation analysis contribute to a deeper understanding of linear systems with these matrices.
• This work offers a foundation for further research in the numerical analysis of structured matrices.