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In this chapter we introduce adjoints of linear maps. In some sense these represent generalizations of the (Hermitian) transposes of a matrices. A matrix is symmetric (or Hermitian) if it is equal to its (Hermitian) transpose. In an analogous way, an endomorphism is selfadjoint if it is equal to its adjoint endomorphism. The sets of symmetric (or Hermitian) matrices and of selfadjoint endomorphisms form certain vector spaces which will play a key role in our proof of the Fundamental Theorem of Algebra in Chap. 15 . Special properties of selfadjoint endomorphisms will be studied in Chap. 18 .

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Adjoints of Linear Maps

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  • 发表日期：

    2025年

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  • Jörg Liesen,
  • Volker Mehrmann

摘要

In this chapter we introduce adjoints of linear maps. In some sense these represent generalizations of the (Hermitian) transposes of a matrices. A matrix is symmetric (or Hermitian) if it is equal to its (Hermitian) transpose. In an analogous way, an endomorphism is selfadjoint if it is equal to its adjoint endomorphism. The sets of symmetric (or Hermitian) matrices and of selfadjoint endomorphisms form certain vector spaces which will play a key role in our proof of the Fundamental Theorem of Algebra in Chap. 15 . Special properties of selfadjoint endomorphisms will be studied in Chap. 18 .