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Engineering

A Note on Full-Rank Factorization of Matrix

Profile image of Prof. Gyan Bahadur Thapa, PhDProf. Gyan Bahadur Thapa, PhD

2019, Journal of the Institute of Engineering

Abstract

We exhibit that the Singular Value Decomposition of a matrix implies a natural full-rank factorization of the matrix

FAQs

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What role does SVD play in full-rank factorization of matrices?add

The study reveals that the Singular Value Decomposition naturally provides a full-rank factorization for any real matrix, ensuring structural integrity across various forms of matrices.

How does the Jordan matrix relate to the eigenvalue problem in SVD?add

Lanczos' analysis indicates that the Jordan matrix facilitates the understanding of the eigenvalue problem, particularly emphasizing real proper values due to the symmetric nature of matrix S.

What are the implications of zero eigenvalues in full-rank factorization?add

The findings show that zero eigenvalues can notably affect the uniqueness of the solutions to linear systems associated with A, especially when the rank p equals n or m.

What distinguishes 'essential axes' in the context of these matrices?add

Essential axes, defined as the proper vectors of symmetric matrix S, are crucial since they correspond to positive eigenvalues necessary for constructing relevant factorization matrices.

How does the constructed factorization ensure uniqueness in linear system solutions?add

The factorization indicates that information from non-null eigenvalues is sufficient to study the existence and uniqueness of linear solutions, allowing for a complete analysis of the matrix A.

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References (14)

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