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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Numerical range ： ウィキペディア英語版 Numerical range In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex ''n'' × ''n'' matrix ''A'' is the set :$W(A)\; =\; \backslash left\backslash \}\}\; \backslash mid\; \backslash mathbf\backslash in\backslash mathbb^n,\backslash \; x\backslash not=0\backslash right\backslash \}$ where x * denotes the conjugate transpose of the vector x. In engineering, numerical ranges are used as a rough estimate of eigenvalues of ''A''. Recently, generalizations of numerical range are used to study quantum computing. A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e. :$r(A)\; =\; \backslash sup\; \backslash \; =\; \backslash sup\_\; |\backslash langle\; Ax,\; x\; \backslash rangle|.$ ''r''(''A'') is a norm. ''r''(''A'')=||''A''||, where ||''A''|| is the operator norm of ''A''. ==Properties== # The numerical range is the range of the Rayleigh quotient. # (Hausdorff–Toeplitz theorem) The numerical range is convex and compact. # $W(\backslash alpha\; A+\backslash beta\; I)=\backslash alpha\; W(A)+\backslash$ for all square matrix ''A'' and complex numbers α and β. Here ''I'' is the identity matrix. # $W(A)$ is a subset of the closed right half-plane if and only if $A+A^$ * is positive semidefinite. # The numerical range $W(\backslash cdot)$ is the only function on the set of square matrices that satisfies (2), (3) and (4). # (Sub-additive) $W(A+B)\backslash subseteq\; W(A)+W(B)$. # $W(A)$ contains all the eigenvalues of ''A''. # The numerical range of a 2×2 matrix is an elliptical disk. # $W(A)$ is a real line segment (β ) if and only if ''A'' is a Hermitian matrix with its smallest and the largest eigenvalues being α and β # If ''A'' is a normal matrix then $W(A)$ is the convex hull of its eigenvalues. # If α is a sharp point on the boundary of $W(A)$, then α is a normal eigenvalue of ''A''. # $r(\backslash cdot)$ is a norm on the space of ''n''×''n'' matrices. # $r(A^n)\; \backslash le\; r(A)^n$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Numerical range」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.