Some functions preserving positive semidefiniteness of 2 × 2 block matrices

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• 作者：Lutz Klotz ^{klotz@math.uni-leipzig.de" class="auth_mail" title="E-mail the corresponding author} ; Conrad Mä ; dler ; ^{maedler@math.uni-leipzig.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15B57 ; 15A30 ; 47B65 ; 30D20
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：507
• 期：Complete
• 页码：68-76
• 全文大小：293 K

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Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si1.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=68afdbab74ef306fbe5169bdb4ff33b6" title="Click to view the MathML source">n∈Nclass="mathContainer hidden">class="mathCode">$n\in \mathbb{N}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si2.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=12fa6b5803cb9c90042dea4dbd230a63" title="Click to view the MathML source">[A_jk]_j,k=1,2class="mathContainer hidden">class="mathCode">${[A_{jk}]}_{j,k=1,2}$ be a Hermitian class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si3.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=68d16c7c3b23fe58b8bd465225b2ed3c" title="Click to view the MathML source">2n×2nclass="mathContainer hidden">class="mathCode">$2n\times 2n$ matrix partitioned into four quadratic matrices class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si4.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=593b14198741b273dd50d9afc7512834" title="Click to view the MathML source">A_jkclass="mathContainer hidden">class="mathCode">$A_{jk}$ of order n  . Marcus and Watkins proved in 1971 that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si5.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=330090572a621aa1e6307d9384747692">class="imgLazyJSB inlineImage" height="21" width="49" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S002437951630204X-si5.gif">class="mathContainer hidden">class="mathCode">$[\mathrm{tr}{A}_{jk}^2]$ is positive semidefinite whenever class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si13.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=94b3944a325a60b33f1e70298054ed28" title="Click to view the MathML source">[A_jk]class="mathContainer hidden">class="mathCode">$[A_{jk}]$ is positive semidefinite. Let A   be an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si7.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=1928b5eecfc4131eae0378fd7ca7d23d" title="Click to view the MathML source">n×nclass="mathContainer hidden">class="mathCode">$n\times n$ matrix. If f   is a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si8.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=f7be242e8afe27fd4d5dc051b70b14f3" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathbb{C}$-valued function on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si8.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=f7be242e8afe27fd4d5dc051b70b14f3" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathbb{C}$, denote by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si9.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=9a50d7ca091e7f5641191ce44abdc60b" title="Click to view the MathML source">f(A)class="mathContainer hidden">class="mathCode">$f(A)$ the value of the primary matrix function associated with f on the matrix A. If F   is a symmetric class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si8.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=f7be242e8afe27fd4d5dc051b70b14f3" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathbb{C}$-valued function on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si10.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=585aaba47536dd5d5ea1ed5959beeb71" title="Click to view the MathML source">Cⁿclass="mathContainer hidden">class="mathCode">${\mathbb{C}}^n$, let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si11.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=627502b9bfa99a3798d7a3578ba11bfb" title="Click to view the MathML source">F(A)class="mathContainer hidden">class="mathCode">$F(A)$ be the value of F on the eigenvalues of A. Generalizing Marcus and Watkins' result we describe those functions f and F  , for which class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si12.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=0946dac7115447a80b1b691e61a74fec" title="Click to view the MathML source">[F(f(A_jk))]class="mathContainer hidden">class="mathCode">$[F(f(A_{jk}))]$ is positive semidefinite whenever class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si13.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=94b3944a325a60b33f1e70298054ed28" title="Click to view the MathML source">[A_jk]class="mathContainer hidden">class="mathCode">$[A_{jk}]$ is positive semidefinite. We extend an example by Choudhury, which gives a negative result for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630204X&_mathId=si14.gif&_user=111111111&_pii=S002437951630204X&_rdoc=1&_issn=00243795&md5=6a3ffeda890a1cb5712070d1ce215631" title="Click to view the MathML source">3×3class="mathContainer hidden">class="mathCode">$3\times 3$ block matrices.