Improving the condition number of a simple eigenvalue by a rank one matrix

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• 作者：Rafael Bru p>rbru@mat.upv.es" class="auth_mail" title="E-mail the corresponding authorp> ; Rafael Cantó ; p> ; p>p>rcanto@mat.upv.es" class="auth_mail" title="E-mail the corresponding authorp> ; Ana M. Urbano p>amurbano@mat.upv.es" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：Brauer&rsquo ; s theorem ; Diagonalization matrix ; Eigenvalue condition number ; Rank-one update
• 刊名：Applied Mathematics Letters
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：58
• 期：Complete
• 页码：7-12
• 全文大小：501 K

文摘

In this work a technique to improve the condition number pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si1.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=877e7060db375558aebf7610c8373470" title="Click to view the MathML source">sb>ib>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> of a simple eigenvalue pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si2.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=44e783e5d49e8da0f1cdd23dd9b0f798" title="Click to view the MathML source">λb>ib>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> of a matrix pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si3.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=dc943246163594cc2b65058e81f4b6cc" title="Click to view the MathML source">A∈Cp>n×np>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> is given. This technique obtains a rank one updated matrix that is similar to pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si4.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=0f6dc5b37c3f2039e1942332216c9397" title="Click to view the MathML source">Apan>pan class="mathContainer hidden">pan class="mathCode">$A$pan>pan>pan> with the eigenvalue condition number of pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si2.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=44e783e5d49e8da0f1cdd23dd9b0f798" title="Click to view the MathML source">λb>ib>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> equal to one. More precisely, the similar updated matrix pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si6.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=a014f1011d0d99b0df3696a86c9bc3d3" title="Click to view the MathML source">A+vb>ib>qp>∗p>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>, where pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si7.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=85c669ff6a84888454bcf08737e3f1cd" title="Click to view the MathML source">Avb>ib>=λb>ib>vb>ib>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> and pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si8.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=4aadc7172789c622ade80cd5fd347e1b" title="Click to view the MathML source">qpan>pan class="mathContainer hidden">pan class="mathCode">$q$pan>pan>pan> is a fixed vector, has pan id="mmlsi9" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si9.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=c53cf85fc153028cb30f9cddb8d6600f" title="Click to view the MathML source">sb>ib>=1pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si8.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=4aadc7172789c622ade80cd5fd347e1b" title="Click to view the MathML source">qpan>pan class="mathContainer hidden">pan class="mathCode">$q$pan>pan>pan>, using only the eigenvalue pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si2.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=44e783e5d49e8da0f1cdd23dd9b0f798" title="Click to view the MathML source">λb>ib>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> and its eigenvector pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300222&_mathId=si12.gif&_user=111111111&_pii=S0893965916300222&_rdoc=1&_issn=08939659&md5=57b055cc54956de54951d082227fa3c6" title="Click to view the MathML source">vb>ib>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>, is given.