On maps preserving products equal to a diagonalizable matrix. (English) Zbl 1478.15040
The authors prove the following theorem, which is an example of a linear preserver problem (see, e.g., [C.-K. Li and N.-K. Tsing, Linear Algebra Appl. 162–164, 217–235 (1992; Zbl 0762.15016)]).
Let \(D_{1}\) and \(D_{2}\) be similar diagonalizable matrices in the matrix algebra \(M_{n}(\mathbb{C})\). Let \(f\) be a bijective linear transformation from \(M_{n}(\mathbb{C})\) onto itself with the property that for all \(A,B\in M_{n}(\mathbb{C})\): \( AB=D_{1}\) \(\implies \) \(f(A)f(B)=D_{2}\). Then it is proved that:
(i) If \(D_{1}\) is singular then there exists a nonzero scalar \(b\) and an invertible matrix \(P\) such that \(b^{2}PD_{1}=D_{2}P\) and \( f(X)=bPXP^{-1}\) for all \(X\in M_{n}(\mathbb{C})\) (*);
(ii) if \(D_{1}\) is nonsingular then either (*) holds or there exists a nonzero scalar \(b\) and invertible matrix \(P\) such that \(b^{2}PD_{1}^{T}=D_{2}^{-1}P\) and \( f(X)=bD_{2}PX^{T}P^{-1}\) for all \(X\in M_{n}(\mathbb{C})\). Conversely, it easy to verify that when \(f\) has these forms \(AB=D_{1}\) \(\implies \) \( f(A)f(B)=D_{2}\).
Let \(D_{1}\) and \(D_{2}\) be similar diagonalizable matrices in the matrix algebra \(M_{n}(\mathbb{C})\). Let \(f\) be a bijective linear transformation from \(M_{n}(\mathbb{C})\) onto itself with the property that for all \(A,B\in M_{n}(\mathbb{C})\): \( AB=D_{1}\) \(\implies \) \(f(A)f(B)=D_{2}\). Then it is proved that:
(i) If \(D_{1}\) is singular then there exists a nonzero scalar \(b\) and an invertible matrix \(P\) such that \(b^{2}PD_{1}=D_{2}P\) and \( f(X)=bPXP^{-1}\) for all \(X\in M_{n}(\mathbb{C})\) (*);
(ii) if \(D_{1}\) is nonsingular then either (*) holds or there exists a nonzero scalar \(b\) and invertible matrix \(P\) such that \(b^{2}PD_{1}^{T}=D_{2}^{-1}P\) and \( f(X)=bD_{2}PX^{T}P^{-1}\) for all \(X\in M_{n}(\mathbb{C})\). Conversely, it easy to verify that when \(f\) has these forms \(AB=D_{1}\) \(\implies \) \( f(A)f(B)=D_{2}\).
Reviewer: John D. Dixon (Ottawa)
MSC:
| 15A86 | Linear preserver problems |
| 15A04 | Linear transformations, semilinear transformations |
| 15A20 | Diagonalization, Jordan forms |
| 15A23 | Factorization of matrices |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
Citations:
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