EVD & SVD EVD & SVD. Durgesh Kumar. June 10, 2016
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1 EVD & SVD Durgesh Kumar June 10, 2016
2 Table of contents 1 Why bother so much for SVD and EVD? 2 Pre-requisite Fundamental of Matrix Multiplication Eigen Value and Eigen Vector 3 EVD Defintion of EVD Use of SVD 4 SVD 5 References
3 Why bother so much for SVD and EVD? Application of SVD and EVD Data projection - Principal Component Analysis. Recommendation engines, e.g., for recommendation systems (such as the Netflix challenge) Page Rank Algorithm ( used for ranking Googles search results ) Data quantization - spectral clustering methods Signal processing, Cryptography, Image/Video/ Speech processing, Pattern recognition, Control theory etc.
4 Pre-requisite Fundamental of Matrix Multiplication Matrix times a Column gives a column Example = A X B 1 1 B is equal to 1* * C1 C 3 B is a Linear combination of columns of A
5 Pre-requisite Fundamental of Matrix Multiplication Matrix times a Column gives a column Example = A X B 1 0 B is equal to 1* * C1 C 2 B is a Linear combination of columns of A
6 Pre-requisite Fundamental of Matrix Multiplication Row times a Matrix gives a row [ 1 ] = [ ] X B A B is equal to R 1 of A + 2* R 2 B is a Linear combination of rows of A
7 Pre-requisite Eigen Value and Eigen Vector Eigen Value and Eigen Vector Defintion v =[1, 0] T, λ = 4 [ ] [ ] [ ] = A v v
8 Pre-requisite Eigen Value and Eigen Vector Eigen Value and Eigen Vector Defintion [ ] [ ] [ ] = A v v v =[1, 0] T, λ = 4 Eigen vectors represent unique direction of information content
9 Pre-requisite Eigen Value and Eigen Vector Eigen Value and Eigen Vector Defintion [ ] [ ] [ ] = A v v v =[1, 0] T, λ = 4 Eigen vectors represent unique direction of information content Eigen value represent the relative weightage of information content in that direction. 1 Play with Eigen Value & Eigen Vector here 2 Online Eigen Values and Eigen Vector Calculation
10 Pre-requisite Eigen Value and Eigen Vector Eigen Value and Eigen Vector Psuedo Example Let s say we want to project a human Say, We have four directtion of projection : forward, backward, top, bottom respectively represting four eigen vectors The ratio of information content in these four direction will be represented by four respective eigen value. Eigen Value in the forwrd direction will be maximum capturing maximum identity of human.
11 EVD Defintion of EVD Eigen Value Decomposition Av1 = λ 1 v1 = Av1 =v1 λ 1 Extending the above aproach, we can write : λ v1 v2... vn v1 v2... vn A... =... 0 λ λ n V V 1 v 1,v 2..., v n are the eiegen vectors of A. 2 λ 1, λ 2,..., λ n are the eigen values of A. Convince yourself about correctness of above equation using Matrix multiplication
12 EVD Defintion of EVD Eigen Value Decomposition Any square matrix A can be written as : A V = V Columns of V represent eigen vectors of A is diagonal matrix conatining eigen values. When A is non-singular matrix, then A = V V 1
13 EVD Defintion of EVD Eigen Value Decomposition Continued When A is non-singular matrix, then A = V V 1 When A is positive-semi definite, then V 1 = V T Hence, A = V V T
14 EVD Defintion of EVD Positive Semi definite matrix Definition 1 : A is called a +ve semi-definite, if A can be expressed as A = XX T Definition 2 : A matrix is said to be +ve semi-definite if we observe following for any non zero vector x, X T A x 0 non-zero x.
15 EVD Defintion of EVD Properties of semi-definite matrix Eigen values are alwayz positive or null. Eigen vectors are pairwise orthogonal, if their eigen values are different. Eigen vector composed of real value.
16 EVD Defintion of EVD Examples of Eigen Value Decomposition A = [ ] Find it s eigen values, eigen vector and eigen value decomposition λ 1 = 3, v 1 =[1, 1] T λ 1 = 3, v 1 =[1, 1] T [ ] [ ] [ ] /2 1/2 A = /2 1/2 V V 1 Why V 1 is not equal to V T, even if A is symmetric?
17 EVD Defintion of EVD Interesting Facts about power of A using EVD If A = V V 1. Then A 2 = V 2 V 1 Eigen vectors of A, A 2,A 3..., are same. Eigen value of A 2 is the sqaure of eigen values of A. If A is an incidence matrix of a grpagh G, then A 2 [i][j] denotes the no of path of length 2 between vertex i to vertex j.
18 EVD Defintion of EVD Diagonalisation of A A = V V 1 This implies = V 1 A V is a diagonal matrix
19 EVD Use of SVD Low rank Aproximation : PCA
20 References References SVD MIT Tutorial SVD Tutorial from very Basic Definition and Explanation of EVD CMU Book Chapter for SVD Intuitive Explanationof SVD Intuitive Explanation of SVD 2
21 References References I
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