Euclidian Algebra
Elements of Mathematics for Economists
Bernard Cornet January 18, 2011
Contents
Notation 1 Euclidean Spaces 1.1 Scalar Product and Associated Norm . . . . . . . . . . . . 1.1.1 Scalar Product . . . . . . . . . . . . . . . . . . . . 1.1.2 Norm Associated to a Scalar Product . . . . . . . . 1.1.3 Convergence in a Normed Space . . . . . . . . . . . 1.1.4 Euclidean Spaces and Hilbert Spaces . . . . . . . . 1.2 Matrices and Scalar Product . . . . . . . . . . . . . . . . . 1.2.1 Generalities on Matrices . . . . . . . . . . . . . . . 1.2.2 Matrices and Scalar Product . . . . . . . . . . . . . 1.2.3 Positive (Negative) Definite Matrices . . . . . . . . 1.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Orthogonal Vectors . . . . . . . . . . . . . . . . . . 1.3.2 Orthogonal Space . . . . . . . . . . . . . . . . . . . 1.3.3 Gram-Schmidt Orthogonalization Process . . . . . 1.4 Orthogonal Projectors . . . . . . . . . . . . . . . . . . . . 1.4.1 Linear Projectors . . . . . . . . . . . . . . . . . . . 1.4.2 Orthogonal Projections . . . . . . . . . . . . . . . . 1.4.3 Symmetric Endomorphisms and Matrices . . . . . 1.4.4 Gram Matrix of a Family of Vectors . . . . . . . . . 1.4.5 Orthogonal Projections and Matrices . . . . . . . . 1.4.6 Least Squares Problem . . . . . . . . . . . . . . . . 1.4.7 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . 1.5 Symmetric Matrices and Endomorphisms . . . . . . . . . . 1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Diagonalization of a Symmetric Matrix . . . . . . . 1.5.3 Positive and Negative Definite Matrices, continued 1 3 5 5 5 7 8 9 10 10 14 15 16 16 17 18 19 19 20 21 23 24 25 25 28 28 28 29
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CONTENTS 1.6 Dual of a Euclidean Space . . . . . . . . . . . . . . . . . 1.6.1 Dual of a Linear Space . . . . . . . . . . . . . . . 1.6.2 Representation of the Dual Space . . . . . . . . .
Bernard Cornet January 18, 2011
Contents
Notation 1 Euclidean Spaces 1.1 Scalar Product and Associated Norm . . . . . . . . . . . . 1.1.1 Scalar Product . . . . . . . . . . . . . . . . . . . . 1.1.2 Norm Associated to a Scalar Product . . . . . . . . 1.1.3 Convergence in a Normed Space . . . . . . . . . . . 1.1.4 Euclidean Spaces and Hilbert Spaces . . . . . . . . 1.2 Matrices and Scalar Product . . . . . . . . . . . . . . . . . 1.2.1 Generalities on Matrices . . . . . . . . . . . . . . . 1.2.2 Matrices and Scalar Product . . . . . . . . . . . . . 1.2.3 Positive (Negative) Definite Matrices . . . . . . . . 1.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Orthogonal Vectors . . . . . . . . . . . . . . . . . . 1.3.2 Orthogonal Space . . . . . . . . . . . . . . . . . . . 1.3.3 Gram-Schmidt Orthogonalization Process . . . . . 1.4 Orthogonal Projectors . . . . . . . . . . . . . . . . . . . . 1.4.1 Linear Projectors . . . . . . . . . . . . . . . . . . . 1.4.2 Orthogonal Projections . . . . . . . . . . . . . . . . 1.4.3 Symmetric Endomorphisms and Matrices . . . . . 1.4.4 Gram Matrix of a Family of Vectors . . . . . . . . . 1.4.5 Orthogonal Projections and Matrices . . . . . . . . 1.4.6 Least Squares Problem . . . . . . . . . . . . . . . . 1.4.7 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . 1.5 Symmetric Matrices and Endomorphisms . . . . . . . . . . 1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Diagonalization of a Symmetric Matrix . . . . . . . 1.5.3 Positive and Negative Definite Matrices, continued 1 3 5 5 5 7 8 9 10 10 14 15 16 16 17 18 19 19 20 21 23 24 25 25 28 28 28 29
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CONTENTS 1.6 Dual of a Euclidean Space . . . . . . . . . . . . . . . . . 1.6.1 Dual of a Linear Space . . . . . . . . . . . . . . . 1.6.2 Representation of the Dual Space . . . . . . . . .