Phase Factors in Singular Value Decomposition and Schmidt Decomposition

• URL: http://arxiv.org/abs/2203.12579v1
• Date: Wed, 23 Mar 2022 17:41:18 GMT
• Title: Phase Factors in Singular Value Decomposition and Schmidt Decomposition
• Authors: Chu Ryang Wie
• Abstract summary: In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AAdag and AdagA are unique up to complex phase factors. We summarize here three simple steps to consistently carry out the SVD and the Schmidt decomposition including the phase factors.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA{\dag} and A{\dag}A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistently carry out the SVD and the Schmidt decomposition including the phase factors.

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