By Nicholas J. Higham
The single booklet dedicated solely to matrix features, this study monograph offers a radical remedy of the speculation of matrix capabilities and numerical tools for computing them. the writer s based presentation makes a speciality of the an identical definitions of f(A) through the Jordan canonical shape, polynomial interpolation, and the Cauchy vital formulation, and contours an emphasis on result of sensible curiosity and an intensive number of difficulties and options. capabilities of Matrices: thought and Computation is greater than only a monograph on matrix features; its wide-ranging content material together with an outline of purposes, ancient references, and miscellaneous effects, methods, and strategies with an f(A) connection makes it important as a normal reference in numerical linear algebra. different key good points of the e-book contain improvement of the idea of conditioning and homes of the Fr?©chet spinoff; an emphasis at the Schur decomposition, the block Parlett recurrence, and sensible use of Pad?© approximants; the inclusion of recent, unpublished study effects and greater algorithms; a bankruptcy dedicated to the f(A)b challenge; and a MATLAB?® toolbox delivering implementations of the major algorithms. viewers: This ebook is for experts in numerical research and utilized linear algebra in addition to an individual wishing to profit in regards to the thought of matrix capabilities and cutting-edge tools for computing them. it may be used for a graduate-level direction on capabilities of matrices and is an acceptable reference for a complicated path on utilized or numerical linear algebra. it's also really well matched for self-study. Contents: record of Figures; record of Tables; Preface; bankruptcy 1: concept of Matrix capabilities; bankruptcy 2: functions; bankruptcy three: Conditioning; bankruptcy four: recommendations for basic services; bankruptcy five: Matrix signal functionality; bankruptcy 6: Matrix sq. Root; bankruptcy 7: Matrix pth Root; bankruptcy eight: The Polar Decomposition; bankruptcy nine: Schur-Parlett set of rules; bankruptcy 10: Matrix Exponential; bankruptcy eleven: Matrix Logarithm; bankruptcy 12: Matrix Cosine and Sine; bankruptcy thirteen: functionality of Matrix instances Vector: f(A)b; bankruptcy 14: Miscellany; Appendix A: Notation; Appendix B: history: Definitions and important proof; Appendix C: Operation Counts; Appendix D: Matrix functionality Toolbox; Appendix E: options to difficulties; Bibliography; Index.
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Extra info for Functions of Matrices: Theory and Computation
See also Drazin, Dungey, and Gruenberg [164, ] for a more elementary proof and the discussions of Taussky [564, ], [565, ]. A complete treatment of simultaneous triangularization is given in the book by Radjavi and Rosenthal [483, ]. Problems The only way to learn mathematics is to do mathematics. — PAUL R. 1. 2 is independent of the particular Jordan canonical form that is used. 2. 2b). Show that f (−λk ) f (−Jk ) = −f ′ (−λk ) f (−λk ) ... . . f (mk −1) (−λk ) (mk − 1)!
Part (a) is immediate from any of the deﬁnitions of h(A). For part (b), let p and q interpolate f and g on the spectrum of A, so that p(A) = f (A) and q(A) = g(A). By diﬀerentiating and using the product rule we ﬁnd that the functions h(t) and r(t) = p(t)q(t) have the same values on the spectrum of A. Hence h(A) = r(A) = p(A)q(A) = f (A)g(A). The next result generalizes the previous one and says that scalar functional relationships of a polynomial nature are preserved by matrix functions. For example sin2 (A) + cos2 (A) = I, (A1/p )p = A, and eiA = cos(A) + i sin(A).
Cullen [125, , Thm. 9]) Deﬁne f (A) by the Jordan canonical form deﬁnition. 38 and the property f (XAX −1 ) = Xf (A)X −1 to show that f (A) is a polynomial in A. 4. (a) Let A ∈ Cn×n have an eigenvalue λ and corresponding eigenvector x. Show that (f (λ), x) is a corresponding eigenpair for f (A). (b) Suppose A has constant row sums α, that is, Ae = αe, where e = [1, 1, . . , 1]T . Show that f (A) has row sums f (α). Deduce the corresponding result for column sums. 5. Show that the minimal polynomial ψ of A ∈ Cn×n exists, is unique, and has degree at most n.