Titlebook: Linear Algebra; Jörg Liesen,Volker Mehrmann Textbook 2015 Springer Nature Switzerland AG 2015 Linear Algebra.Matrices.Echelon Form.Gaussia

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Linear Forms and Bilinear Forms,self. These maps play an important role in many areas of Mathematics, including Analysis, Functional Analysis and the solution of differential equations. They will also be essential for the further developments in this book: Using bilinear and sesquilinear forms, which are introduced in this chapter

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Euclidean and Unitary Vector Spaces,on such vector spaces. Scalar products allow the extension of well-known concepts from elementary geometry, such as length and angles, to abstract real and complex vector spaces. This, in particular, leads to the idea of orthogonality and to orthonormal bases of vector spaces. As an example for the

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Adjoints of Linear Maps,rix 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

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Polynomials and the Fundamental Theorem of Algebra,ncluding the factorization into irreducible factors. We also prove the Fundamental Theorem of Algebra, which states that every non-constant polynomial over the complex numbers has a least one complex root. This implies that every complex matrix and every endomorphism on a (finite dimensional) comple

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Cyclic Subspaces, Duality and the Jordan Canonical Form,icularly interested in the algebraic and geometric multiplicities of the eigenvalues of . and the characterization of the corresponding eigenspaces. Our strategy in this analysis is to decompose the vector space . into a direct sum of .-invariant subspaces so that, with appropriately chosen bases, t

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