Numerical Range of Moore–Penrose Inverse Matrices

Numerical Range of Moore–Penrose Inverse Matrices

Let <i>A</i> be an <i>n</i>-by-<i>n</i> matrix. The numerical range of <i>A</i> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <m...

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Main Author: Mao-Ting Chien
Format: Article
Language:English
Published: MDPI AG 2020-05-01
Series:Mathematics
Subjects:
Moore–Penrose inverse
numerical range
weighted shift matrix
Online Access:https://www.mdpi.com/2227-7390/8/5/830
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author Mao-Ting Chien
author_facet Mao-Ting Chien
author_sort Mao-Ting Chien
collection DOAJ
description Let <i>A</i> be an <i>n</i>-by-<i>n</i> matrix. The numerical range of <i>A</i> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mi>A</mi> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. The Moore–Penrose inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>+</mo> </msup> </semantics> </math> </inline-formula> of <i>A</i> is the unique matrix satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo>,</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>,</mo> <msup> <mrow> <mo>(</mo> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> </mrow> </semantics> </math> </inline-formula>. This paper investigates the numerical range of the Moore–Penrose inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>+</mo> </msup> </semantics> </math> </inline-formula> of a matrix <i>A</i>, and examines the relation between the numerical ranges <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>.
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spelling doaj.art-ef5c946ae71d4e88913615edf7c7c38f2023-11-20T01:06:36ZengMDPI AGMathematics2227-73902020-05-018583010.3390/math8050830Numerical Range of Moore–Penrose Inverse MatricesMao-Ting Chien0Department of Mathematics, Soochow University, Taipei 111002, TaiwanLet <i>A</i> be an <i>n</i>-by-<i>n</i> matrix. The numerical range of <i>A</i> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mi>A</mi> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. The Moore–Penrose inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>+</mo> </msup> </semantics> </math> </inline-formula> of <i>A</i> is the unique matrix satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo>,</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>,</mo> <msup> <mrow> <mo>(</mo> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> </mrow> </semantics> </math> </inline-formula>. This paper investigates the numerical range of the Moore–Penrose inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>+</mo> </msup> </semantics> </math> </inline-formula> of a matrix <i>A</i>, and examines the relation between the numerical ranges <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/8/5/830Moore–Penrose inversenumerical rangeweighted shift matrix
spellingShingle Mao-Ting Chien
Numerical Range of Moore–Penrose Inverse Matrices
Mathematics
Moore–Penrose inverse
numerical range
weighted shift matrix
title Numerical Range of Moore–Penrose Inverse Matrices
title_full Numerical Range of Moore–Penrose Inverse Matrices
title_fullStr Numerical Range of Moore–Penrose Inverse Matrices
title_full_unstemmed Numerical Range of Moore–Penrose Inverse Matrices
title_short Numerical Range of Moore–Penrose Inverse Matrices
title_sort numerical range of moore penrose inverse matrices
topic Moore–Penrose inverse
numerical range
weighted shift matrix
url https://www.mdpi.com/2227-7390/8/5/830
work_keys_str_mv AT maotingchien numericalrangeofmoorepenroseinversematrices

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