CMSE 820: Math. Foundations of Data Sci.

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1 Lecture Weighted path graphs Take from [10, Lecture 3] As alluded to at the ed of the previous sectio, we ow aalyze weighted path graphs. To that ed, we prove the followig: Theorem 6 (Fiedler). Let P =(V, E, w) be a weighted path graph o vertices, let L P have eigevalues 0 = l 1 < l 2 apple applel, ad let j k be a eigevector with eigevalue l k. The j k chages sig k 1 times. We will eed to first prove a few lemmas i order to prove Theorem 6. The first of these is Sylvester s Law of Iertia: Theorem 7 (Sylvester s Law of Itertia). Let A be ay symmetric matrix ad let B be ay o-sigular matrix (that is, B has o zero sigular values). The, the matrix BAB T has the same umber of positive, egative ad zero eigevalues as A. Proof. We first recall three facts from liear algebra. 1. The first is that BAB 1 has the same eigevalues as A, sice: Aj = lj () BAB 1 (Bj) =l(bj). 2. The secod fact is that rak(a) =rak(bab). 3. The third is that every osigular matrix B ca be writte B = QR, where Q is a orthoormal matrix (meaig Q T Q = QQ T = I) ad R is a upper-triagular matrix with positive diagoals (this is the so-called QR factorizatio). We are goig to begi by usig a slight variatio of the last fact, ad write B = RQ. Now, sice Q T = Q 1, by the first fact we kow that QAQ T has exactly the same eigevalues as A. Defie 8 t 2 [0, 1], R t = tr +(1 t)i, 88

2 ad cosider the family of matrices 8 t 2 [0, 1], M t = R t QAQ T R T t. At t = 0 we have R 0 = I ad so M 0 = QAQ T has the same eigevalues as A. For t = 1 we have M 1 = BAB T. Sice all of the matrices M t are symmetric, they all have real eigevalues (by the Spectral Theorem). Additioally, the eigevalues of a symmetric matrix are cotiuous fuctios of the etries of the matrix. Therefore, if the umber of positive, egative, or zero eigevalues of BAB T differs from that of A, the there must be some t for which M t has more zero eigevalues tha does A. But the matrices R t are upper triagular with positive diagoal etries, ad hece are o-sigular (sice the eigevalues of R t are the diagoal etries). Thus the rak of M t must equal the rak of A, which meas this caot happe. Fiedler s Theorem will follow from a aalysis of the eigevalues of tri-diagoal matrices with zero row-sums. These may be viewed as Laplacias of weighted path graphs i which some edges are allowed to have egative weights. Lemma 2. Let M be a symmetric matrix such that M1 = 0. The: M = M ij L Gi,j. (53) Proof. Equatio (53) is a equality betwee two matrices. Let A deote the right had side matrix. O the off diagoal it is clear that both M (the LHS) ad A (the RHS) are equal. Notice as well that the right had side satisfies: A1 = M ij L Gi,j 1 = M ij 0 = 0. Thus M1 = 0 ad A1 = 0. Notice that these are sets of equatios ad ukows (i.e., the diagoal etries), which have uique solutios. Sice the off diagoal etries of M ad A are idetical, the equatios are the same, ad thus the solutios are as well, meaig that the diagoal of M ad A are the same. 89

3 Lemma 3. Let M be a symmetric tri-diagoal matrix with 2q positive off-diagoal etries such that M1 = 0. The M has q egative eigevalues. Proof. By Lemma 2 ad the fact that M is symmetric ad tri-diagoal, we may write: Thus for v 2 R, v T Mv = M = M i 1,i L Gi 1,i. M i 1,i (v[i 1] v[i]) 2. Now we perform a chage variables that will diagoalize the matrix M. Let d[1] =v[1] ad set d[i] =v[i] v[i 1] for i 2, so that: v[i] =d[1]+d[2]+ + d[i]. Notice that if we defie the lower triagular matrix T as: T = B A, the v = Td. By Sylvester s Law of Iertia (Theorem 7), we kow that A = T T MT has the same umber of positive, egative ad zero eigevalues as M. O the other had, d T Ad = d T T T MTd = v T Mv = M i 1,i d[i] 2. Thus A has oe zero eigevalue (with eigevector d[1] =1, d[j] =0 for all j 2) ad a egative eigevalue M i 1,i for each M i 1,i > 0 (with eigevector d[i] =1, d[j] =0 for all j 6= i), of which there are q. 90

4 Proof of Theorem 6. We cosider the case whe j k has o zero etries. The proof for the geeral case may be obtaied by splittig the graph by removig the vertices with zero etries. For simplicity, we also assume that l k has multiplicity 1. Recall we wish to show that j k chages sig k 1 times. This is equivalet to showig that: #{i = 1,..., 1:j k [i]j k [i + 1] < 0} = k 1. Let V k deote the diagoal matrix with j k o the diagoal. Cosider the matrix: M = Vk T (L P l k I)V k. The ier matrix L P l k I has oe zero eigevalue ad k 1 egative eigevalues derived from the eigevalues ad eigevectors of L P. So, by Sylvester s Law of Iertia (Theorem 7), M has k 1 egative eigevalues, oe zero eigevalue, ad k postitive eigevalues. We are ow goig to use Lemma 3. The matrix M is clearly symmetric ad tri-diagoal, ad additioally: M1 = V T k (L P l k I)V k 1 = V T k (L P l k I)j k = V T k 0 = 0. Thus we ca apply Lemma 3 to M. We ote additioally that M i,i+1 = w(i, i + 1)j k [i]j k [i + 1], ad thus we see that M i,i+1 is positive precisely whe j k [i]j k [i + 1] < 0. Sice M has k 1 egative eigevalues, by Lemma 3 it must have k 1 positive etries o the upper diagoal, which meas that j k [i]j k [i + 1] < 0 must occur for exactly k 1 idices. 91

5 Refereces [1] Berhard Schölkopf ad Alexader J. Smola. Learig with Kerels: Support Vector Machies, Regularizatio, Optimizatio, ad Beyod. Adaptive Computatio ad Machie Learig. The MIT Press, [2] Afoso S. Badeira. Te lectures ad forty-two ope problems i the mathematics of data sciece. MIT course Topics i Mathematics of Data Sciece, [3] Jo Shles. A tutorial o pricipal compoet aalysis. arxiv: , [4] Karl Pearso. O lies ad plaes of closest fit to systems of poits i space. Philosophical Magazie, Series 6, 2(11): , [5] V. A. Marcheko ad L. A. Pastur. Distributio of eigevalues i certai sets of radom matrices. Mat. Sb. (N.S.), 72(114): , [6] J. Baik, G. Be-Arous, ad S. Péché. Phase trasitio of the largest eigevalue for oull complex sample covariace matrices. The Aals of Probability, 33(5): , [7] Debashis Paul. Asymptotics of sample eigestructure for a large dimesioal spiked covariace model. Statistica Siica, pages , [8] Trevor Hastie, Robert Tibshirai, ad Jerome Friedma. The Elemets of Statistical Learig. Spriger-Verlag New York, 2d editio, [9] Athaasios Tsaas ad Ageliki Xifara. Accurate quatitative estimatio of eergy performace of residetial buildigs usig statistical machie learig tools. Eergy ad Buildigs, 49: , [10] Daiel A. Spielma. Spectral graph theory. Yale Course Notes, Fall,