1381170045_768__232_lecture_4.4

3 25 ix4 13 30 ix5 14 dec 10 random structures and

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Unformatted text preview: arni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 It is a subspace because it the solution set of a homogeneous linear system. It consists of the eigenvectors associated with λ along with the zero vector (subspaces always contain 0) C EDRIC C HAUVE , FALL 2013 7 f a cu lty of science d epa r tm ent of m athema tic s Week Date Sections Part/ References MATH 895-4 Fall 2010 Course Schedule Topic/Sections MATH 232 S ECTION #4.4 Notes/Speaker from FS2009 Summary: Finding Eigenvalues and Eigenspaces To find the eigen1 Sept 7 I.1, I.2, I.3 Symbolic methods Combinatorial vectors ofI.5, I.6square a Structures 2 14 I.4, Unlabelled structures 3 4 5 21 II.1, II.2, II.3 28 II.4, II.5, II.6 FS: Part A.1, A.2 Comtet74 Handout #1 (self study) Labelled structures I 1. Find the characteristic polynomial p(λ) = det(λI − A) of A Labelled structures II Combinatorial Asst #1 Due 2. Oct 5 III.1,the roots of this Combinatorial Find III.2 polynomial: these are the eigenvalues parameters Parameters 6 12 9 Nov 2 IV.1, IV.2 FS A.III (self-study) Multivariable GFs 3. 19ForIV.3, IV.4 eigenvalue λComplex Analysis each , the eigenspace corresponding to it is null(λI − A): 7 Analytic Methods this is the FS: Part of all eigenvectors for λ plus the zero vector set B: IV, V, VI Singularity Analysis 8 26 Appendix B4 10 11 12 IV.5 V.1 Stanley 99: Ch. 6 Handout #1 (self-study) Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial instances of discrete Mariolys Random Structures and Limit Laws FS: Part C (rotating presentations) Sophie 25 IX.4 Continuous Limit Laws The still expanding fundamentalMarni Theorem (4.4.7) If A is an n × n maQuasi-Powers and 13 30 IX.5 Sophie Gaussian limit laws trix, then the following statements are equivalent 14 Dec 10 Presentations Asst #3 Due (a). The reduced row echelon form of A is In . (b). A is expressible as a product of elementary matrices. (c). A is invertible. (d). Ax = 0 has only the trivial solution. Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 (e). Ax = b is consistent for every b ∈ Rn . (f). Ax = b has exactly one solution for every vector b ∈ Rn . (g). The columns of A are linearly independent. (h). The rows of A are linearly independent. (i). det(A) = 0. (j). λ = 0 is not an eigenvalue of A. C EDRIC C HAUVE , FALL 2013 8 MATH 895-4 Fall 2010 Course Schedule f a cu lty of science d epa r tm ent of m athema tic s Week Date 1 Sept 7 Sections Part/ References Topic/Sections MATH 232 S ECTION #4.4 Notes/Speaker from FS2009 Eigenvalues/Eigenvectors of a Power of a Matrix I.1, I.2, I.3 Symbolic methods Combinatorial SupposeI.4, I.5, I.6 AStructuressquare matrix with eigenvalue λ that FS: Part A.1, A.2 is a 2 14 Unlabelled structures 3 21 II.1, II.2, II.3 Comtet74 Handout #1 (self study) Labelled structures I Labelled structures A And suppose x is an eigenvector of II corresponding to λ 4 28 II.4, II.5, II.6 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 7 19 IV.3, IV.4 8 26 9 Nov 2 IV.5 V.1 Combinatorial parameters FS A.III (self-study) Combinatorial Parameters Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis Asst #1 Due Multivariable GFs Singularity Analysis Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 Quasi-Powers...
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