Matrix differential calculus with applications in statistics and econometrics
A brand new, fully updated edition of a popular classic on matrix differential calculus with applications in statistics and econometrics This exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Hoboken, NJ
John Wiley & Sons, Inc.
2019
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| Edition: | Third edition |
| Series: | Wiley series in probability and statistics
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| Subjects: | |
| Online Access: | |
| Collection: | O'Reilly - Collection details see MPG.ReNa |
Table of Contents:
- Part Two
- Differentials: the theoryChapter 4 Mathematical preliminaries; 1 Introduction; 2 Interior points and accumulation points; 3 Open and closed sets; 4 The Bolzano-Weierstrass theorem; 5 Functions; 6 The limit of a function; 7 Continuous functions and compactness; 8 Convex sets; 9 Convex and concave functions; Bibliographical notes; Chapter 5 Differentials and differentiability; 1 Introduction; 2 Continuity; 3 Differentiability and linear approximation; 4 The differential of a vector function; 5 Uniqueness of the differential; 6 Continuity of differentiable functions
- 18 Positive (semi)definite matrices19 Three further results for positive definite matrices; 20 A useful result; 21 Symmetric matrix functions; Miscellaneous exercises; Bibliographical notes; Chapter 2 Kronecker products, vec operator, and Moore-Penrose inverse; 1 Introduction; 2 The Kronecker product; 3 Eigenvalues of a Kronecker product; 4 The vec operator; 5 The Moore-Penrose (MP) inverse; 6 Existence and uniqueness of the MP inverse; 7 Some properties of the MP inverse; 8 Further properties; 9 The solution of linear equation systems; Miscellaneous exercises; Bibliographical notes
- 7 Partial derivatives8 The first identification theorem; 9 Existence of the differential, I; 10 Existence of the differential, II; 11 Continuous differentiability; 12 The chain rule; 13 Cauchy invariance; 14 The mean-value theorem for real-valued functions; 15 Differentiable matrix functions; 16 Some remarks on notation; 17 Complex differentiation; Miscellaneous exercises; Bibliographical notes; Chapter 6 The second differential; 1 Introduction; 2 Second-order partial derivatives; 3 The Hessian matrix; 4 Twice differentiability and second-order approximation, I
- Chapter 3 Miscellaneous matrix results1 Introduction; 2 The adjoint matrix; 3 Proof of Theorem 3.1; 4 Bordered determinants; 5 The matrix equation AX = 0; 6 The Hadamard product; 7 The commutation matrix Kmn; 8 The duplication matrix Dn; 9 Relationship between Dn+1 and Dn, I; 10 Relationship between Dn+1 and Dn, II; 11 Conditions for a quadratic form to be positive (negative) subject to linear constraints; 12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B); 13 The bordered Gramian matrix; 14 The equations X1A + X2B′ = G1,X1B = G2; Miscellaneous exercises; Bibliographical notes
- Includes bibliographical references and indexes
- Cover; Title Page; Copyright; Contents; Preface; Part One
- Matrices; Chapter 1 Basic properties of vectors and matrices; 1 Introduction; 2 Sets; 3 Matrices: addition and multiplication; 4 The transpose of a matrix; 5 Square matrices; 6 Linear forms and quadratic forms; 7 The rank of a matrix; 8 The inverse; 9 The determinant; 10 The trace; 11 Partitioned matrices; 12 Complex matrices; 13 Eigenvalues and eigenvectors; 14 Schur's decomposition theorem; 15 The Jordan decomposition; 16 The singular-value decomposition; 17 Further results concerning eigenvalues