%%% (*) Problem 1 due Monday; Problems 2-6 due Please hand in 2-6 on Monday if possible. Problems 2-6 due on Friday, September 20. (*) September 16-22 is British Columbia Reconciliation Week at the PNE; classes suspended Wednesday, September 18. "I encourage everyone to participate." -- Professor Stephen Toope, President and Vice-Chancellor, UBC. (*) Continue to diagonalize [1/2,1/3;1/2,2/3] %%% Today: Diagonalize [1/2,1/3;1/2,2/3] (HW 1) you should make sure that you can do this with any reasonable 2 x 2 matrix. For an algorithm, see 3,000 Solved ..., Section 16.3. Theorem: Markov matrix has 1 as eigenvalue. see Subsections 4.6.1 and 4.6.2. see Stochastic Matrix, wikipedia (note they use row vectors, not column vectors) Theorem: Sum of eigenvalues = Trace see Trace (linear algebra), wikipedia %%% "From the late 19th century, until the end of the 20th century, the government of Canada, in collaboration with church groups, ran a system of Indian Residential Schools. Aboriginal children were often forcibly removed from their families at a very young age, and many spent their entire childhood in these schools, where they suffered emotional, physical, and sexual abuse. Many died there." -- Dr. Linc Kesler, Director, UBC First Nations House of Learning, Our Truth UBC - September 18, http://www.youtube.com/watch?v=xjXM5SSzJ-s&feature=youtu.be %%% On September 18, UBC will suspend classes in Vancouver so we can all connect to the powerful testimonies heard at the Truth and Reconciliation Commision. "I encourage everyone to participate." -- Professor Stephen Toope, President and Vice-Chancellor, UBC, Our Truth UBC - September 18, http://www.youtube.com/watch?v=xjXM5SSzJ-s&feature=youtu.be %%% Consider going to the PNE on Wednesday, Sept 18, to participate in the TRC. Feast on salmon and bannock! Free admission! http://www.trc.ca "If a man dwells on the past, then he robs the present; but if a man ignores the past, he may rob the future." - Master Po [Keye Luke], "Kung Fu" [TV program] http://www.youtube.com/watch?v=IKH4nY4WlFY %%% Back to our regular program... [ 1/2 1/3 ] [ ] eigenvalues lambda = 1, 1/6 [ 1/2 2/3 ] lambda = 1 [ -1/2 1/3 ] A - lambda I = [ ] [ 1/2 -1/3 ] [ -1/2 1/3 ] [ v_1 ] [ 0 ] [ ] [ ] = [ ] [ 1/2 -1/3 ] [ v_2 ] [ 0 ] (1/2) v_1 = (1/3) v_2 ; [ v_1; v_2] = [ 3/2; 1 ] or [ 30/2; 10 ] or [-3; -2 ] or [ .6 ; .4 ] or ... %%% Back to our regular program... [ 1/2 1/3 ] [ ] eigenvalues lambda = 1, 1/6 [ 1/2 2/3 ] lambda = 1/6 [ 1/3 1/3 ] A - lambda I = [ ] [ 1/2 1/2 ] [ 1/3 1/3 ] [ v_1 ] [ 0 ] [ ] [ ] = [ ] [ 1/2 1/2 ] [ v_2 ] [ 0 ] v_1 = - v_2 ; [ v_1; v_2] = [ -1 ; 1 ] or [ 10 ; -10 ] or [-3; 3 ] or [ -.7 ; .7 ] or ...