Mappings preserving spectra of products of matrices. (English) Zbl 1113.15002
Let\(M_{n}\) be the set of all \(n\times{n}\) complex matrices. It is a well-known result that if a linear map \(\varphi:M_{n}\rightarrow{M_{n}}\) preserves the eigenvalues (counting multiplicities) of each matrix in \(M_{n}\) then \(\varphi\) should be either in the form \(\varphi(A)=S^{-1}AS\) or \(\varphi(A)=S^{-1}A^{t}S\), for some invertible matrix \(S\) in \(M_{n}\). Some generalizations of this result have been developed [L. Molnár, Proc. Am. Math. Soc. 130, No. 1, 111–120 (2002; Zbl 0983.47024) and M. Omladic and P. Semrl, Linear Algebra Appl. 153, 67–72 (1991; Zbl 0736.47001)].
The authors in the paper under review consider a more challenging generalization of the above mentioned result. Let \({k}\geq{2}\) and let \(\{j_{1},\dots ,j_{m}\}\) be a sequence of integers such that \(\{j_{1},\dots ,j_{m}\}=\left\{{1,\dots ,k}\right\}\). Suppose there is \(j_{r}\) not equal to \(j_{s}\) for all \(s\neq{r}\). Now define \(X_{1}*\dots *X_{k}\)=\(X_{j_{1}}\dots X_{j_{m}}\); where the product on the right hand side is the usual product of matrices.
The authors prove the following theorem in several steps: A mapping \(\varphi:M_{n}\rightarrow{M_{n}}\)satisfies \(Sp(\varphi(X_{1})*\dots *\varphi(X_{k}))=Sp(X_{1}*\dots *X_{k})\) for all \({X_{i}}\in{M_{n}}\) if and only if there exist a non-singular matrix \(S\) in \(M_{n}\) and a complex number \(\xi\) satisfying \(\xi^{m}={1}\) such that \(\varphi\) has the form \(\varphi(A)={\xi}S^{-1}AS\) or \(\varphi(A)={\xi}S^{-1}A^{t}S\).
They extend their results to the Jordan product \(A*B=(AB+BA)/2\) by considering \(X_{1}*\dots *X_{k}\)=\(\left(X_{j_{1}}\dots X_{j_{m}}+X_{j_{m}}\dots X_{j_{1}}\right)/2\). They obtain similar results for Hermitian and real symmetric matrices as well.
The authors in the paper under review consider a more challenging generalization of the above mentioned result. Let \({k}\geq{2}\) and let \(\{j_{1},\dots ,j_{m}\}\) be a sequence of integers such that \(\{j_{1},\dots ,j_{m}\}=\left\{{1,\dots ,k}\right\}\). Suppose there is \(j_{r}\) not equal to \(j_{s}\) for all \(s\neq{r}\). Now define \(X_{1}*\dots *X_{k}\)=\(X_{j_{1}}\dots X_{j_{m}}\); where the product on the right hand side is the usual product of matrices.
The authors prove the following theorem in several steps: A mapping \(\varphi:M_{n}\rightarrow{M_{n}}\)satisfies \(Sp(\varphi(X_{1})*\dots *\varphi(X_{k}))=Sp(X_{1}*\dots *X_{k})\) for all \({X_{i}}\in{M_{n}}\) if and only if there exist a non-singular matrix \(S\) in \(M_{n}\) and a complex number \(\xi\) satisfying \(\xi^{m}={1}\) such that \(\varphi\) has the form \(\varphi(A)={\xi}S^{-1}AS\) or \(\varphi(A)={\xi}S^{-1}A^{t}S\).
They extend their results to the Jordan product \(A*B=(AB+BA)/2\) by considering \(X_{1}*\dots *X_{k}\)=\(\left(X_{j_{1}}\dots X_{j_{m}}+X_{j_{m}}\dots X_{j_{1}}\right)/2\). They obtain similar results for Hermitian and real symmetric matrices as well.
Reviewer: Manouchehr Misaghian (Charlotte)
MSC:
| 15A04 | Linear transformations, semilinear transformations |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Keywords:
eigenvalue; spectrum; preserving map; Hermitian matrices; real symmetric matrices; Jordan productReferences:
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