Unformatted text preview: arni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 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 8954 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
(selfstudy) 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
(selfstudy) 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 maQuasiPowers 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: 11Dec09 (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 8954 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
(selfstudy) Combinatorial
Parameters Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) 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 QuasiPowers...
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 Fall '10
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 Math, Linear Algebra, Algebra, Eigenvalue, eigenvector and eigenspace, Diagonal matrix, Triangular matrix, Dr. Marni MISHNA

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