Computing matrix symmetrizers, part 2: New methods using eigendata and linear means; a comparison

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• 作者：Froilá ; n Dopico^a ; ^{dopico@math.uc3m.es" class="auth_mail" title="E-mail the corresponding author} ; Frank Uhlig^b ; ^{uhligfd@auburn.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65F10 ; 65J10 ; 15A23 ; 15B57
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 September 2016
• 年：2016
• 卷：504
• 期：Complete
• 页码：590-622
• 全文大小：608 K

文摘

Over any field lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515003845&_mathId=si1.gif&_user=111111111&_pii=S0024379515003845&_rdoc=1&_issn=00243795&md5=ac7e41eddb4f2e86264b55cc582e2693" title="Click to view the MathML source">Flass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">F every square matrix A   can be factored into the product of two symmetric matrices as lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515003845&_mathId=si2.gif&_user=111111111&_pii=S0024379515003845&_rdoc=1&_issn=00243795&md5=cee3b499324b150c78075cad05b6340e" title="Click to view the MathML source">A=S₁⋅S₂lass="mathContainer hidden">lass="mathCode">low="scroll">A=S1⋅S2 with lsi3" class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515003845&_mathId=si3.gif&_user=111111111&_pii=S0024379515003845&_rdoc=1&_issn=00243795&md5=132954aedc93c465bd200994e4162ade">lass="imgLazyJSB inlineImage" height="18" width="109" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515003845-si3.gif">lass="mathContainer hidden">lass="mathCode">low="scroll">Si=SiT∈le-struck">Fn,n and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910.

Frobenius' symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013  and . The resulting iterative algorithm has solved this computational problem over lsi4" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515003845&_mathId=si4.gif&_user=111111111&_pii=S0024379515003845&_rdoc=1&_issn=00243795&md5=0374f82d57b76365334aea70c50348bf" title="Click to view the MathML source">Rlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">R and lsi5" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515003845&_mathId=si5.gif&_user=111111111&_pii=S0024379515003845&_rdoc=1&_issn=00243795&md5=5514f36b1cdba9a5b3b61e5752e88000" title="Click to view the MathML source">Class="mathContainer hidden">lass="mathCode">low="scroll">le-struck">C, but at high computational cost.

This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram–Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta's 1973 method that symmetrizes unreduced Hessenberg matrices directly.

The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization.