New PDF release: Matrix Differential Calculus with Applications in Statistics

By Jan R. Magnus

ISBN-10: 047198633X

ISBN-13: 9780471986331

Matrix Differential Calculus With purposes in data and Econometrics Revised version Jan R. Magnus, middle, Tilburg collage, The Netherlands and Heinz Neudecker, Cesaro, Schagen, The Netherlands " .deals carefully with some of the difficulties that experience bedevilled the topic as much as the current time." - Stephen Pollock, Econometric conception "I persevered to be pleasantly stunned by means of the range and usability of its contents " - Isabella Verdinelli, magazine of the yank Statistical organization carrying on with the luck in their first version, Magnus and Neudecker current an exhaustive and self-contained revised textual content on matrix idea and matrix differential calculus. Matrix calculus has turn into a necessary instrument for quantitative tools in lots of functions, starting from social and behavioural sciences to econometrics. whereas the constitution and winning parts of the 1st version stay, this revised and up to date variation includes many new examples and exercises.
* comprises the necessities of multivariable calculus with an emphasis at the use of differentials
* Many new examples and exercises
* Fulfils the necessity for a unified and self-contained remedy of matrix differential calculus
* contains new advancements during this field
half I provides a concise, but thorough assessment of matrix algebra, whereas the second one half develops the idea of differentials. the remainder components III to VI mix the speculation and alertness of matrix differential calculus offering the practitioner and researcher with either a short overview and a close reference. stopover at our website

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Additional info for Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Edition

Sample text

The equality (4) can be generalized as follows. Theorem 3 Let A, B, C and D be four matrices such that the matrix product ABCD is defined and square. Then, tr ABCD = (vec D′ )′ (C ′ ⊗ A) vec B = (vec D)′ (A ⊗ C ′ ) vec B ′ . (9) 36 Kronecker products, the vec operator and the Moore-Penrose inverse [Ch. 2 Proof. We have, using (4) and (5), tr ABCD = tr D(ABC) = (vec D′ )′ vec ABC = (vec D′ )′ (C ′ ⊗ A) vec B. (10) The second equality is proved in the same way starting from tr ABCD = tr D′ (C ′ B ′ A′ ).

SCHUR’S DECOMPOSITION THEOREM In the next few sections we present three decomposition theorems: Schur’s theorem, Jordan’s theorem and the singular-value decomposition. Each of these theorems will prove useful later in this book. We first state Schur’s theorem. Theorem 12 (Schur decomposition) Let A be an n × n matrix. Then there exist a unitary n × n matrix S (that is, S ∗ S = In ) and an upper triangular matrix M whose diagonal elements are the eigenvalues of A, such that S ∗ AS = M. (1) The most important special case of Schur’s decomposition theorem is the case where A is symmetric.

Proof (uniqueness). Assume that two matrices B and C both satisfy the four defining conditions. Then AB = (AB)′ = B ′ A′ = B ′ (ACA)′ = B ′ A′ C ′ A′ = (AB)′ (AC)′ = ABAC = AC. (1) Similarly, BA = (BA)′ = A′ B ′ = (ACA)′ B ′ = A′ C ′ A′ B ′ = (CA)′ (BA)′ = CABA = CA. (2) B = BAB = BAC = CAC = C. (3) Hence, Proof (existence). Let A be an m × n matrix with r(A) = r. If r = 0, then A = 0 and A+ = 0 satisfies the four defining equations. Assume therefore r > 0. 16 there exist semi-orthogonal matrices S and T and a positive definite diagonal r × r matrix Λ such that A = SΛ1/2 T ′ , S ′ S = T ′ T = Ir .

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Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Edition by Jan R. Magnus

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