PPT-Graphs, Vectors, and Matrices

Daniel A Spielman Yale University AMS Josiah Willard Gibbs Lecture January 6 2016 From Applied to Pure Mathematics Algebraic and Spectral Graph Theory Sparsification a pproximating graphs by graphs with fewer edges. Ther e ar e many applications of linear algebra for example chemists might use ow eduction to get a lear er pictur e of what elements go into a complicated eaction In this lectur e we explor e the linear algebra associated with electrical networks G The following are equivalent is PSD ie Ax for all all eigenvalues of are nonnegative for some real matrix Corollary Let be a homogeneous quadratic polynomial Then for all if and only if for some Rudi Pendavingh TUE Semide64257nite matrices Con Such matrices has several attractive properties they support algorithms with low computational complexity and make it easy to perform in cremental updates to signals We discuss applications to several areas including compressive sensing data stream Sometimes, two graphs have exactly the same form, in the sense that there is a one-to-one correspondence between their vertex sets that preserves edges. In such a case, we say that the two graphs are . Nickolay. . Balonin. . and . Jennifer . Seberry. To Hadi. for your 70. th. birthday. Spot the Difference!. Mathon. C46. Balonin. -Seberry C46. In this presentation. Two Circulant Matrices. Two Border Two Circulant Matrices. Honors Advanced Algebra II/Trigonometry. Ms. . lee. Essential. Stuff. Essential Question: What is a matrix, and how do we perform mathematical operations on matrices?. Essential Vocabulary:. Matrix. Section . 10.3. Representing Graphs: . Adjacency Lists. Definition. : An . adjacency list . can be used to represent a graph with no multiple edges by specifying the vertices that are adjacent to each vertex of the graph.. Matrix Multiplication. Matrix multiplication is defined differently than matrix addition. The matrices need not be of the same dimension. Multiplication of the elements will involve both multiplication and addition. A . matrix. . M. is an array of . cell entries. (. m. row,column. ) . that have . rectangular. . dimensions. (. Rows x Columns. ).. Example:. 3x4. 3. 4. 15. x. Dimensions:. A. a. row,column. A. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: . 29. th. August 2015. Introduction. A matrix (plural: matrices) is . simply an ‘array’ of numbers. , e.g.. But the power of matrices comes from being able to multiply matrices by vectors and matrices by matrices and ‘invert’ them: we can:. What is a matrix?. A Matrix is just rectangular arrays of items. A typical . matrix . is . a rectangular array of numbers arranged in rows and columns.. Sizing a matrix. By convention matrices are “sized” using the number of rows (m) by number of columns (n).. Objectives: to represent translations and dilations w/ matrices. : to represent reflections and rotations with matrices. Objectives. Translations & Dilations w/ Matrices. Reflections & Rotations w/ Matrices. K-means. Input: set of data points, k. Randomly pick k points as means. For . i. in [0, . maxiters. ]:. Assign each point to nearest center. Re-estimate each center as mean of points assigned to it. Rotation of coordinates -the rotation matrixStokes Parameters and unpolarizedlight1916 -20041819 -1903Hans Mueller1900 -1965yyxyEEEElinear arbitrary anglepolarization right or left circularpolarizati