Last week David talked about this paper http://arxiv.org/abs/1010.2921 Today I talked about the following graph isomorphism paper: |

For the third and last HW assignment in this course please submit by the end of the semester problem 1 in problem set 3 and problems 1-3 in problem set 4 on Dan's course website: http://www.cs.yale.edu/homes/spielman/561/ |

In previous recent meetings Eli Discussed efficient approximation for linear equations, following lectures 15-16 of Spielman. The lecture this Tuesday Dec 21st was based on the following paper We continue reviewing this paper next Tuesday. p.s.- will probably return ex. 1 next Tuesday |

The material we covered is the second half of Lecture 2 and first half of Lecture 3 in Spielman's notes. -(note that below I use C(i,j) for the (i,j)'th entry in the matrix C, as opposed to class where I used subscripts) I stated this as *exactly* k times, but as was noted in class this does not make sense when the eigenvalue has multiplicity>1. So what is correct is the `at most' statement, and the `exactly' statement is true only when the k'th smallest eigenvalue has multiplicity one. 2. I had a problem with the proof of the above theorem at the following point: We had a symmetric matrix C such that for any vector x x^T*C*x = \sum_i lambda_i *(x_i)^2. I wanted to say that this implies the lambda_i's are eigenvalues of C. This is true since in the bilinear form x^T*C*x, for a symmetric C, the coefficient of x_i*x_j is 2*C(i,j) , so the equality above implies C is a diagonal matrix with C(i,i) =lambda_i in the diagonal. What I claimed about certain unit vectors being eigenvectors was not true. |

Eli covered Lecture 5 in the notes. |

I discussed the Perron-Frobenius Theorem. A simpler proof in my opinion for positive matrices, without subtle analysis points, can be found here http://www.stanford.edu/class/ee363/lectures/pf.pdf starting from slide 19. Eli followed Lectures 12-13 from a different course by Spielman, this one: http://www.cs.yale.edu/homes/spielman/462/ The first homework set, due in two weeks (Nov 16) is Problem set 1 from Dan's course, here's a link: http://www.cs.yale.edu/homes/spielman/561/ps1-09.pdf |