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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Linear transformations on matrices


Author: D. Ž. Djoković
Journal: Trans. Amer. Math. Soc. 198 (1974), 93-106
MSC: Primary 20G20
DOI: https://doi.org/10.1090/S0002-9947-1974-0374285-3
MathSciNet review: 0374285
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Abstract: The real orthogonal group $ O(n)$, the unitary group $ U(n)$ and the symplectic group $ {\text{sp(}}n{\text{)}}$ are embedded in a standard way in the real vector space of $ n \times n$ real, complex and quaternionic matrices, respectively. Let $ F$ be a nonsingular real linear transformation of the ambient space of matrices such that $ F(G) \subset G$ where $ G$ is one of the groups mentioned above. Then we show that either $ F(x) = a\sigma (x)b$ or $ F(x) = a\sigma ({x^\ast })b$ where $ a,b \in G$ are fixed, $ {x^\ast }$ is the transpose conjugate of the matrix $ x$ and $ \sigma $ is an automorphism of reals, complexes and quaternions, respectively.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0374285-3
Keywords: Orthogonal, unitary and symplectic group, invariant subspace, tangent space, skew-hermitian matrices, Noether-Skolem theorem
Article copyright: © Copyright 1974 American Mathematical Society