The Spectral Basis of a Linear Operator

Transcription

1 The Spectral Basis of a Linear Operator Garret Sobczyk Universidad de Las Americas-P Cholula, Mexico Thursday Jan. 10, 2013, 2PM AMS/MAA Joint Math Meeting San Diego Convention Center

2 Abstract The idea of a spectral basis first arises in modular or clock arithmetic but is an even more powerful tool in linear algebra and numerical analysis. The spectral basis of a linear operator, uniquely determined by its minimal polynomial, exhibits the macro structure of a linear operator in terms of the basic building blocks of mutually annihilating idempotents and nilpotents which determine its generalized eigenspaces. These ideas are only part of a much larger program developed by the author in his new book, "New Foundations in Mathematics: The Geometric Concept of Number" (Birkhauser 2013), which uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Starting with linear algebra, geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering.

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4 What is Geometric Algebra? Geometric algebra is the completion of the real number system to include new anticommuting square roots of plus and minus one, each such root representing an orthogonal direction in successively higher dimensions.

5 Contents I. Beyond the Real Numbers. a) Clock arithmetic. b) Modular polynomials and approximation. b) Complex numbers. c) Hyperbolic numbers. II. III. IV. The Geometric Concept of Number. Linear Algebra and Matrices. a) Matrices of geometric numbers. b) Geometric numbers and determinants. c) The spectral decomposition. Splitting Space and Time.

6 Contents V. Geometric Calculus. VI. VII. Differential Geometry. Non-Euclidean and Projective Geometries a) The affine plane b) Projective geometry c) Conics d) The horosphere VIII. Lie groups and Lie algebras. a) Bivector representation b) The general linear group c) Orthogonal Lie groups and algebras d) Semisimple Lie Algebras IX. Conclusions X. Selected References

7 Clock Arithmetic 12 = 3x2 2 Spectral equation: s 1 + s 2 = 1 or 3(s 1 + s 2 ) = 3 s 2 = 3. This implies that 9 s 2 = s 2 = 9, and s 1 = 4. Now define q 2 = 2 s 2 = 6. Spectral basis: { s 1, s 2, q 2 } idempotents: s 1 2 = 16 = 4 mod 12 = s 1 s 2 2 = 81 = 9 mod 12 = s 2 nilpotent: q 2 2 = 36 = 0 mod 12, s 1 s 2 = 0 mod 12

8 Clock Arithmetic: 12 = 3x2 2 A calculation: 5s 1 + 5s 2 = 5mod(12) or 2s 1 + 1s 2 = 5 mod(12). It follows that 2 n s n s 2 = 5 n mod(12) for all integers n. n=-1 gives 1/5 = 2s 1 + 1s 2 = 5 mod(12) and n=100 gives = s 1 + s 2 = 1 mod(12).

9 Modular Polynomials and Interpolation mod(h(x))

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11 Complex and Hyperpolic Numbers u 2 =1

12 Hyperbolic Numbers

13 : Geometric Numbers G 2 of the Plane Standard Basis of G 2 ={1, e 1, e 2, e 12 }. where i=e 12 is a unit bivector.

14 ab =a. b+a^b a. b=½(ab+ba) a^b=½(ab-ba) a 2 =a. a= a 2 Basic Identities

15 Geometric Numbers of 3-Space a^b=i axb a^b^c=[a. (bxc)]i where i=e 1 e 2 e 3 =e 123 a. (b^c)=(a. b)c-(a. c)b = - ax(bxc).

16 Reflections L(x) and Rotations R(x) where a = b =1 and

17 Matrices of the Geometric Algebra G 2 Recall that G 2 =span{1, e 1, e 2, e 12 }. By the spectral basis of G 2 we mean where are mutually annihiliating idempotents. Note that e 1 u + = u - e 1.

18 For example, if then the element g Ɛ G 2 is We find that

19 Matrices of the Geometric Algebra G 3 We can get the complex Pauli matrices from the matrices of G 2 by noting that e 1 e 2 = i e 3 or e 3 = -i e 1 e 2, where i = e 123 is the unit element of volume of G 3. We get

20 Geometric numbers and determinants Let a 1, a 2,..., a n be vectors in R n, where Then

21 Spectral Decomposition Let with the characteristic polynomial φ(x)=(x-1)x 2. Recall that the spectral basis for this polynomial was Replacing x by the matrix X, and 1 by the identity 3x3 matrix gives

22 It follows that the spectral equation for X is X=1 S S 2 + Q 2, with the eigenvectors We now obtain the Jordan Normal Form for X

23 The ordinary rotation Splitting Space and Time is in the blue plane of the bivector i=e 12. The blue plane is boosted into the yellow plane by with the velocity v/c = Tanh ɸ. The light cone is shown in red.

24 Minkowski Space R 1,3 g 0 is timelike, g 1 g 2 g 3 spacelike

25 Spacetime Algebra G 1,3 We start with We factor e 1, e 2, e 3 into Dirac bivectors, where

26 Conformal Mappings Conformal mapping the unit cylinder onto the figure shown.

27 A more exotic conformal mapping of the hyperboloid like figure into the figure surrounding it.

28 Non-Euclidean and Projective The affine plane. Each point x in R n determines a unique point x h in the affine plane. Geometries

29 Desargue s Configuration Thm: Two triangles are in perspective axially if and only if they are in perspective centrally.

30 Any conformal transformation can be represented by an orthogonal transformation on the horosphere. The Horosphere

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32 Lie Algebras and Lie Groups Let G n,n be the 2 2n -dimensional geometric algebra with neutral signature. The Witte basis consists of two dual null cones: We now construct the matrix of bivectors

33 These bivectors are the generators of the general linear Lie algebra gl n, with the Lie bracket product Each bivector F generates a linear transformation f, defined by

34 General Linear Group The general linear group GL n is obtained from the Lie algebra gl n by exponentiation. We have GL n = { G=e F F Ɛ gl n }. Consider now the one parameter subgroups defined for each G Ɛ gl n by g t (x)=e ½tF x e -½tF where x= x i a i and t Ɛ R. Differentiating gives It follows that

35 Conclusions The spectral basis of idempotents and nilpotents define both the eigenvalues and eigenspaces of a linear operator. The spectral basis also has important applications in number theory, numerical analysis and advanced mathematics. Geometric algebra is built upon the geometric concept of number and offers new tools for the study of linear algebra and projective geometry, and provides a unified approach to diverse areas of mathematics, physics and the engineering sciences. I hope my selection of topics has been sufficiently broad to show that geometric algebra and the Geometric Concept of Number provides a New Foundation for Mathematics.

36 Selected References W.K. Clifford, Applications of Grassmann's extensive algebra, Amer. J. of Math. 1 (1878), P. J. Davis, Interpolation and Approximation, Dover Publications, New York, T.F. Havel, GEOMETRIC ALGEBRA: Parallel Processing for the Mind (Nuclear Engineering) D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed., Kluwer D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, 2nd edition, Kluwer G. Sobczyk, The missing spectral basis in algebra and number theory, The American Mathematical Monthly 108 April 2001, pp P. Lounesto, Clifford Algebras and Spinors, 2nd Edition. Cambridge University Press, Cambridge, G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal, 26:4 (1995) G. Sobczyk, The Generalized Spectral Decomposition of a Linear Operator, The College Mathematics Journal, 28:1 (1997) G. Sobczyk, Spacetime vector analysis, Physics Letters, 84A, 45 (1981). J. Pozo and G. Sobczyk. Geometric Algebra in Linear Algebra and Geometry}. Acta Applicandae Mathematicae, 71: , Note: Copies of many of my papers and talks can be found on my website: