<p>Greville’s spectral <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{1,2,3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>-inverses of a square complex matrix <i>A</i> are solutions to the four matrix equations <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$AXA=A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>A</mi> <mo>=</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$XAX=X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mi>A</mi> <mi>X</mi> <mo>=</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(AX)^{*}=AX$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>=</mo> <mi>A</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$XA^{k+1}=A^{k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <msup> <mi>A</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (for some integer <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$k \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>), possessing remarkable least-squares and spectral properties. In this paper, we give a canonical form of spectral <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\{1,2,3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>-inverses under the core-EP decomposition, which suggests a special kind of spectral <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\{1,2,3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>-inverse. We develop the main properties and characterizations of this special generalized inverse and show its applications in minimization problems of some linear systems.</p>

A canonical form of Greville’s spectral $\{1,2,3\}$-inverse with applications

期刊论文
发表日期：

2026年1月5日

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Cang Wu,
Dijana Mosić,
Guiqi Shi

摘要

Greville’s spectral $\{1,2,3\}$ ${1, 2, 3}$ -inverses of a square complex matrix A are solutions to the four matrix equations $AXA=A$ $A X A = A$ , $XAX=X$ $X A X = X$ , $(AX)^{*}=AX$ ${(A X)}^{*} = A X$ and $XA^{k+1}=A^{k}$ $X A^{k + 1} = A^{k}$ (for some integer $k \ge 0$ $k \geq 0$ ), possessing remarkable least-squares and spectral properties. In this paper, we give a canonical form of spectral $\{1,2,3\}$ ${1, 2, 3}$ -inverses under the core-EP decomposition, which suggests a special kind of spectral $\{1,2,3\}$ ${1, 2, 3}$ -inverse. We develop the main properties and characterizations of this special generalized inverse and show its applications in minimization problems of some linear systems.

A canonical form of Greville’s spectral \(\{1,2,3\}\)-inverse with applications

摘要