{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:40:09Z","timestamp":1787323209171,"version":"3.56.0"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1996,7]]},"abstract":"<jats:p>In 1967, Kahan obtained a matrix extension theorem: Suppose $H \\in \\mathbb{C}^{l \\times l} $ is Hermitian and $B \\in \\mathbb{C}^{s \\times l} $. Denote the spectral norm of \\[ R = \\begin{bmatrix} H \\\\ B \\end{bmatrix} \\] by $ \\| R \\|_2 $. Then there exists a $W \\in \\mathbb{C}^{s \\times s} $ such that \\[ A = \\begin{bmatrix} H &amp; {B^* } \\\\ B &amp; W \\end{bmatrix} \\] is Hermitian and $ \\| A \\|_2 = \\| R \\|_2 $. Kahan did not give an explicit expression for W. We show that one may take\\[ ( 1 )\\qquad W = - BH( \\varrho^2 I - H^2 )^\\dag B^ * , \\] where $A^\\dag $ denotes the Moore\u2013Penrose generalized inverse of A. Furthermore, the inequality \\[ ( 2 )\\qquad B ( \\varrho I + H )^\\dag B^* - \\varrho I \\leq W \\leq \\varrho I - B ( \\varrho I - H )^\\dag B^* \\] gives the \u201cgeneral solution formula\u201d for W in Kahan\u2019s theorem, where $A \\geq B$ means A and B are Hermitian and $A - B$ is positive semidefinite. A by-product of (2) is the inequality \\[ ( 3 )\\qquad 2\\varrho I \\geq B [ ( \\varrho I + H )^\\dag + ( \\varrho I - H )^\\dag ] B . \\] In this paper we also consider the following problem: Suppose $H \\in \\mathbb{C}^{l \\times l} $ is normal, $B \\in \\mathbb{C}^{s \\times l} $, and \\[ R = \\begin{bmatrix} H \\\\ B \\end{bmatrix} . \\] How can we find a Hermitian W and a matrix $B_1 $ such that $\\| B_1 \\|_2 = \\| B \\|_2$ and $\\| A \\|_2 = \\| R \\|_2 $, where \\[ A = \\begin{bmatrix} H &amp; {B_1^* } \\\\ B &amp; W \\end{bmatrix}? \\]<\/jats:p>","DOI":"10.1137\/0617037","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"621-631","source":"Crossref","is-referenced-by-count":4,"title":["Further Study and Generalization of Kahan\u2019s Matrix Extension Theorem"],"prefix":"10.1137","volume":"17","author":[{"given":"Dao-Sheng","family":"Zheng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0117041"},{"key":"R2","volume-title":"Generalized inverses: theory and applications","author":"Ben-Israel Adi","year":"1974"},{"key":"R3","volume-title":"Matrix computations","author":"Golub Gene H.","year":"1989"},{"key":"R4","unstructured":"W. M. Kahan,  Inclusion Theorems for Clusters Eigenvalues of Hermitian Matrices, Computer Science report, Univ. of Toronto, Toronto, Canada,  1967"},{"key":"R5","volume-title":"The symmetric eigenvalue problem","author":"Parlett Beresford N.","year":"1980"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0117004"},{"key":"R7","volume-title":"Introduction to matrix computations","author":"Stewart G. W.","year":"1973"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/1019104"},{"key":"R9","volume-title":"Matrix perturbation theory","author":"Stewart G. W.","year":"1990"},{"key":"R10","volume-title":"The algebraic eigenvalue problem","author":"Wilkinson J. H.","year":"1965"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/0617037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:48:15Z","timestamp":1787320095000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/0617037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,7]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["10.1137\/0617037"],"URL":"https:\/\/doi.org\/10.1137\/0617037","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,7]]}}}