Titlebook: Matrices; Theory and Applicati Denis Serre Textbook 2010Latest edition The Editor(s) (if applicable) and The Author(s), under exclusive lic

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Elementary Linear and Multilinear Algebra,This chapter is the only one where results are given either without proof, or with sketchy proofs. A beginner should have a close look at a textbook dedicated to linear algebra, not only reading statements and proofs, but also solving exercises in order to become familiar with all the relevant notions.

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Tensor and Exterior Products,Let . and . be .-vector spaces whose dimensions are finite. We construct their .. ⊗.. as follows.

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Norms,In this chapter, the field .is always .or .and .denotes ... The scalar (if .=.) or Hermitian (if .=.) product on .is denoted by .

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Matrices with Entries in a Principal Ideal Domain; Jordan Reduction,In this chapter we consider only .(see Chapter 3). Such a ring .can be embedded in its field of fractions, which is the quotient of . by the equivalence relation . The embedding is the map ..

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Exponential of a Matrix, Polar Decomposition, and Classical Groups,Polar decomposition and exponentiation are fundamental tools in the theory of finite-dimensional Lie groups and Lie algebras. We do not consider these notions here in their full generality, but restrict attention to their matricial aspects.

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Matrix Factorizations and Their Applications,The techniques described below are often called .

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