By Jan R. Magnus
Matrix Differential Calculus With functions in facts and Econometrics Revised variation Jan R. Magnus, heart, Tilburg college, The Netherlands and Heinz Neudecker, Cesaro, Schagen, The Netherlands " .deals carefully with a number of the difficulties that experience bedevilled the topic as much as the current time." - Stephen Pollock, Econometric idea "I persisted to be pleasantly stunned via the diversity and usability of its contents " - Isabella Verdinelli, magazine of the yank Statistical organization carrying on with the good fortune in their first version, Magnus and Neudecker current an exhaustive and self-contained revised textual content on matrix conception and matrix differential calculus. Matrix calculus has turn into a necessary software for quantitative tools in a lot of functions, starting from social and behavioural sciences to econometrics. whereas the constitution and profitable parts of the 1st version stay, this revised and up to date variation comprises many new examples and exercises.
* includes 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 therapy of matrix differential calculus
* comprises new advancements during this field
half I provides a concise, but thorough evaluate of matrix algebra, whereas the second one half develops the speculation of differentials. the rest elements III to VI mix the idea and alertness of matrix differential calculus delivering the practitioner and researcher with either a short overview and a close reference. stopover at our online page http://www.wiley.com/
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Additional resources for Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Edition
Ix) is established by substitution in the defining equations using (vii) and (viii). (x) follows from (ix) and (vii); (xi) follows from (ix) and (viii); (xii) and (xiii) follow from (xi) and (i). To prove (xv), note that B + A+ = (B ′ B)+ B ′ A′ (AA′ )+ , using (xi). Finally, to prove (xvi) we use (xi) and (x) and write A+ B = 0 ⇐⇒ (A′ A)+ A′ B = 0 ⇐⇒ A′ A(A′ A)+ A′ B = 0 ⇐⇒ A′ B = 0. ✷ Exercises Sec. 8 ] Further properties 39 1. Determine a+ , where a is a column vector. 2. If r(A) = 1, show that A+ = (tr AA′ )−1 A′ .
The ‘vec’ notation was introduced by Koopmans, Rubin and Leipnik (1950). Theorem 2 is due to Roth (1934). §5–§8. The Moore-Penrose inverse was introduced by Moore (1920, 1935) and rediscovered by Penrose (1955). There exists a large amount of literature on generalized inverses, of which the Moore-Penrose inverse is one example. The interested reader may wish to consult Rao and Mitra (1971), Pringle and Rayner (1971), Boullion and Odell (1971), or Ben-Israel and Greville (1974). §9. The results in this section are due to Penrose (1956).
A similar result holds, of course, for AA− . 4. Show that A(A′ A)− A′ = A(A′ A)+ A′ and hence is symmetric and idempotent. 5. Show that a necessary and sufficient condition for the equation Ax = b to have a solution is that AA− b = b, in which case the general solution is x = A− b + (I − A− A)q where q is an arbitrary vector of appropriate order. ) 6. Show that (AB)+ = B + A+ if A has full column rank and B has full row rank. 7. Show that (A′ A)2 B = A′ A if and only if A+ = B ′ A′ . 8. If A and B are positive semidefinite and AB = BA, show that (B 1/2 A+ B 1/2 )+ = B +1/2 AB +1/2 (Liu 1995).
Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Edition by Jan R. Magnus