For a natural number n, denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {M}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> the algebra of all matrices of size n over the complex field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. Let also <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> be a fixed nonzero vector in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {C}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper, we characterize linear bijective maps on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {M}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> sending nilpotent matrices into matrices of local spectral radius zero at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. As a corollary, we also give the general form of linear bijective maps on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {M}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> sending matrices of local spectral radius zero at <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> into matrices of the same type.

Linear Bijective Maps Sending Nilpotent Matrices into Matrices of Local Spectral Radius Zero at a Fixed Vector

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2026年6月18日

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Constantin COSTARA

摘要

For a natural number n, denote by $\mathcal {M}_n$ $M_{n}$ the algebra of all matrices of size n over the complex field $\mathbb {C}$ $C$ . Let also $x_0$ $x_{0}$ be a fixed nonzero vector in $\mathbb {C}^{n}$ $C^{n}$ . In this paper, we characterize linear bijective maps on $\mathcal {M}_n$ $M_{n}$ sending nilpotent matrices into matrices of local spectral radius zero at $x_0$ $x_{0}$ . As a corollary, we also give the general form of linear bijective maps on $\mathcal {M}_n$ $M_{n}$ sending matrices of local spectral radius zero at $x_0$ $x_{0}$ into matrices of the same type.