SPECTRAL THEOREM AND APPLICATIONS

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1 SPECTRAL THEOREM AND APPLICATIONS JINGJING (JENNY) LI Abstract. This paper is dedicated to preset a proof of the Spectral Theorem, ad to discuss how the Spectral Theorem is applied i combiatorics ad graph theory. I this paper, we also give isights ito the ways i which this theorem uveils some mysteries i graph theory, such as expader graphs ad graph colorig. Cotets 1. Itroductio 1 2. Proof of the Spectral Theorem 2 3. Cosequeces ad Applicatios Spectral Graph Theory 3 Ackowledgmets 8 Refereces 8 1. Itroductio The topic of this paper is a fudametal theorem of mathematics: The Spectral Theorem. This theorem cocers symmetric trasformatios o fiite dimesioal Euclidea spaces. It specifies a coditio for matrix diagoalizatio, which is widely used i discrete mathematics ad other fields icludig physical scieces (for example, quatum mechaics). I order to uderstad the Spectral Theorem, first we eed to look at some defiitios: Defiitio 1.1. Let V be a Euclidea vector space. A set of vectors {b 1,..., b k } V forms a orthoormal basis if { 0 if i j, b i, b j = 1 if i = j. ad the set spas V. Defiitio 1.2. Let V be as above ad let φ be a liear trasformatio from V to itself. φ is a symmetric trasformatio if ( x, y V )( φ(x), y = x, φ(y) ). If we represet φ with respect to a orthoormal basis the we get a symmetric matrix, i.e. A ij = A ji. The coverse is also true: if A is a symmetric matrix, the it defies a symmetric trasformatio o R. This justifies the use of the same adjective for both otios. [ ] 1 2 For example, A = defies a symmetric trasformatio o R Date: September 23,

2 2 JINGJING (JENNY) LI Now let us state the Spectral Theorem: Theorem 1.3. I a fiite-dimesioal Euclidea space, every symmetric trasformatio has a orthoormal eigebasis. I this paper, we will prove this theorem i several steps that will be preseted as a series of separate lemmas. We will coclude by icludig specific examples of the theorem s applicatio to graph theory, such as the theory of expader graphs. 2. Proof of the Spectral Theorem To prove the Spectral Theorem, the first step is to prove that there exists oe eigevector. The, we wat to show that assumig the theorem holds true for a subspace, we ca prove the theorem for the orthogoal complemet of that space. Lastly, combiig the first ad secod steps, we ca prove the etire Spectral Theorem by iductio, because we kow there always exists oe eigevector. φv, v Defiitio 2.1. For every v 0, defie the Rayleigh quotiet as v, v. Furthermore, defie φv, v λ = sup v 0 v, v. Note that the fuctio v φv, v is a cotiuous fuctio o the compact set {v v = 1}. Therefore, by the Extreme Value Theorem, the umber λ is attaied. To prove the Spectral Theorem, we eed to fid a eigebasis. So the first step is to fid oe eigevector. The ext lemma will show that a vector that maximizes the Rayleigh quotiet is actually a eigevector. Lemma 2.2. Suppose v 0 = 1 satisfies φv 0, v 0 = λ, the φv 0 = λv 0. Proof. Sice λ is the supremum of all Rayleigh quotiets, for all v ad w such that v + w 0, we have λ φ(v + w), v + w v + w, v + w = λ v + w, v + w φ(v + w), v + w. We otice that this last iequality holds eve if v + w = 0. Expadig both sides of the iequality, we have λ( v, v + w, w + 2 v, w ) φv, v + φw, w + φv, w + φw, v. Sice φ is symmetric, φw, v = w, φv = φv, w, ad thus we get λ v, v + λ w, w + 2 λv, w φv, v + φw, w + 2 φv, w. Sice the iequality holds for ay v ad w, it holds particularly for v = v 0 ad w = tu, where t R ad u V is ay vector. Sice v 0 = 1 ad φv 0, v 0 = λ, the previous iequality becomes λ + λt 2 u, u + 2t λu, u λ + t 2 φu, u + 2t φv 0, u = t 2 (λ u, u φu, u ) + 2t( λv 0 φv 0, u ) 0. As t varies we see that the left-had side of the iequality traces a parabola i t. At t = 0 the parabola vaishes, ad the iequality demostrates that t = 0 is i fact a global miimum, so the derivative at 0 vaishes. Thus, for all u U,

3 SPECTRAL THEOREM AND APPLICATIONS 3 we deduce that λv 0 φv 0, u = 0. I particular for u = λv 0 φv 0, we get 0 = λv 0 φv 0, λv 0 φv 0 = λv 0 φv 0 2. So φv 0 = λv 0. Now that we have oe eigevector, we eed to cotiue this process ad produce additioal eigevectors iside {v 0 }. Lemma 2.3. Let U be a subspace of V. If U is φ-ivariat, the U is also φ-ivariat. Proof. Let u U. We wat to show that φu U, i.e. for every u U, it holds that φu, u = 0. Sice φ is symmetric, u, φu = u, φu = 0, because the φ-ivariace of U implies φu U ad u U. Lemma 2.4. Let V be a Euclidea vector space, ad let U V be a liear subspace with ier product iduced from V. As before, let φ be a symmetric trasformatio. If U is φ-ivariat, the φ U is symmetric. Proof. By defiitio, φu = φ U (u). So for all u, u U, the symmetry of φ implies φu, u V = u, φu V. Sice the ier product o U is the restrictio of the ier product o V, we have φ U (u), u U = u, φ U (u ) U. Therefore φ U is symmetric. With the lemmas prove above, we ca ow proceed to prove the Spectral Theorem. Proof of Theorem 1.3. Base case: dim V = 1. Take ay ozero vector v i this oe-dimesioal vector space. Divide v by its orm so that it becomes a orthoormal basis. The vector φv is a scalar multiple of the give vector v. Iductio hypothesis: for k = 1, a symmetric k k matrix has a orthoormal eigebasis. The sice dim U = 1, U is φ-ivariat. Iductive step: Let λ = max x, φx = v 0, φv 0. By Lemma 2.2, we have φv 0 = x =1 λv 0. Defie U = spa{v 0 } as a φ -ivariat subspace of V. Sice dim U = 1, it follows that dim U = 1. By Lemma 2.3, U is φ-ivariat, ad by Lemma 2.4, φ U is symmetric. By the iductio hypothesis, φ U has orthoormal eigebasis {v 1,..., v 1 } o U. Therefore, we coclude {v 0, v 1,..., v 1 } is a orthoormal eigebasis of φ o V. 3. Cosequeces ad Applicatios Spectral Graph Theory Now, we will look at the specific ways i which the Spectral Theorem ad its cosequeces are applied to solve problems i graph theory, icludig how eigevalues illumiate some mysteries of graphs. We have defied λ to be the supremum of all Rayleigh quotiets. The ext lemma shows that λ is the maximum of all eigevalues. Lemma 3.1. For ay eigevalue λ i of φ, we have λ i λ, where λ = max x, φx. x =1

4 4 JINGJING (JENNY) LI Proof. If λ i is a eigevalue of V ad v i is a correspodig eigevector, without loss of geerality, we may assume that v i = 1. So we deduce λ i = λ i v i, v i = v i, λ i v i = v i, φv i max x, φx = λ. x =1 I the same way that we fid λ, we ca fid all eigevalues. Throughout this sectio, we deote orthoormal eigevectors v 1,..., v, where for all i [], v i has correspodig eigevalue λ i. Additioally, the λ s are decreasig, ad so λ 1 λ 2... λ. Oe of the most widely used cosequeces of the Spectral Theorem is the Courat-Fischer mi-max priciple. Theorem 3.2 (Courat-Fischer Priciple). The eigevalues λ 1 λ of the symmetric matrix A are give by λ k = max mi S R x S,x 0 dim(s)=k x t x = mi S R max x S,x 0 dim(s )= k+1 x t x. Proof. First we prove that λ k is achievable. By the Spectral Theorem, there exists a orthoormal eigebasis for A. Suppose v 1,..., v form such a orthoormal eigebasis. Let S k = spa {v 1,..., v k }. For all x S k, we ca expad ad express x = quotiet x t x = k λ i (v t i x) 2 k (v it x) 2 k (v i t x)v i. Cosiderig the Rayleigh k λ k (v t i x) 2 = λ k. k (v it x) 2 Next we prove that λ k is the maximum over all such expressios. Let S k = spa {v k,..., v }. Sice dim S k = k + 1, for all S such that dim S = k we have S S k. The we ca deduce that mi x S x t x mi x t x. Similar to previous expressios, for all x S S, we ca write x = Sice for all i k we have λ k λ i, it follows that x t λ i (v t i x) 2 λ k (v t i x) 2 Ax x t x = i=k i=k = λ k. (v it x) 2 (v it x) 2 i=k i=k x S S k i=k (v i t x)v i. We have thus prove for all S R with dimesio k that λ k mi x S x t x. The proof for the equality λ k = mi max S R x S x t x is aalogous. dim(s )= k+1 Now, we will look at the specific ways i which the Spectral Theorem ad its cosequeces are applied to solve problems i combiatorics ad graph theory. For

5 SPECTRAL THEOREM AND APPLICATIONS 5 all the followig examples, let G be a fiite graph, whose vertices are labeled 1 through. Example 3.3 (Adjacecy matrix). We defie A to be the adjacecy matrix such that { 1 if i ad j are adjacet, A ij = 0 otherwise. We ote that the matrix A ij is symmetric, ad this will provide the coectio betwee the Spectral Theorem ad its applicatio to graph theory. We ca use adjacecy matrices i a lot of differet ways. For example, deote 0. e i = 1, where 1 is i the i-th place. The [1,..., 1]Ae i = the umber of vertices adjacet to the i-th vertex. We defie this umber to be the degree of the i-th vertex, deoted by deg(v i ). Example 3.4 (Laplacia matrix). The Laplacia matrix is a matrix that averages over the eighbors of every vertex, ad it is defied as follows: Let D be the degree matrix whose i-th etry is equal to the degree of the i-th vertex. The Laplacia matrix of G is defied as L = D A. More formally, deg(v i ) if i = j, L ij = 1 if i j ad v i is adjacet to v j, 0 otherwise. The Laplacia matrix plays a importat role i radom walks o graphs. Defiitio 3.5. We let λ deote the largest absolute value of ay eigevalue other max λ i λ>1 tha the largest oe, that is, λ =. The spectral gap is the differece d betwee the largest eigevalue ad the eigevalue whose absolute value is the secod largest, which is equal to λ 1 λ. We are iterested i the spectral gap because it is related to the radomess of the graph i a subtle way, which we will later see more clearly i the Expader Mixig Lemma. Further, i order to get a quatity that ca be compared across differet graphs, we proceed to ormalize the spectral gap. Defiitio 3.6. A graph is d-regular if all of its vertices have degree d, which represets the regularity ad equals the largest eigevalue. We defie. 0 spectral expasio = spectral gap. d Remark 3.7. Cosider a d-regular graph G = (V, E) whose adjacecy matrix A has eigevalues λ 1 λ 2 λ. We let d = λ 1, which is the largest eigevalue of A. Let S ad T be sets of vertices from G, i.e. S, T V. The umber of possible

6 6 JINGJING (JENNY) LI pairs of vertices from S ad T is S T. The umber of edges betwee pairs of d vertices from S ad T is expected to be S T, because for a arbitrary 1 pair of vertices i, j of G, the probability that there is a edge betwee i ad j is d umber of edges ( = ( 2 = = 2) 2) d 1. d 2 ( 1) 2 Before we state the Expader Mixig Lemma, let us take a look at the followig lemma that cocers the relatioship betwee the largest eigevalue of S ad degree of vertices i the subgraph of S. Lemma 3.8 (Average Degree Lemma). Let S V be a subset of vertices, ad cosider the subgraph of V spaed by S. Let deg ave (S) deote the average degree of vertices i the subgraph S (whose edges are all betwee vertices of S). The deg ave (S) λ 1. Proof. Let v(s) ad v(t ) be the characteristic vectors of S ad T. Cosiderig the Rayleigh quotiet, we ca deduce The we get λ 1 = max x 0 By the defiitio of v(s), we have We kow x t x v(s)t Av(S) v(s) t v(s). v(s) t v(s) = v(s) 2 = S. v(s) t Av(S) = (u,v) S S A(u, v) = u S (u,v) S S A(u, v). A(u, v) = deg(u), because here we are coutig the umber of eighborig vertices of u, which is the defiitio of degree. Let be the size of S. The we get v(s) t v(s) =. I coclusio, λ 1 = max x 0 x t x v(s)t Av(S) v(s) t v(s) = v S (u,v) S S A(u, v) = u S deg(u) = deg ave (S). Oe fial igrediet for the Expader Mixig Lemma is the expader graph. Defiitio 3.9. A λ-expader graph is a d-regular graph whose spectral expasio is 1 λ (which is equal to λ 1 d λ). Ituitively, o a expader graph radom walks coverge quickly to the statioary distributio. The speed of covergece is a cosequece of the Expader Mixig Lemma, which we will prove ext. It is importat to ote that give the size of the spectral expasio, this lemma allows us to kow how close the graph is from beig i the radom situatio.

7 SPECTRAL THEOREM AND APPLICATIONS 7 Theorem 3.10 (Expader Mixig Lemma). Let G = (V, E) be a d-regular, -vertex graph with spectral expasio 1 λ, such that its adjacecy matrix A has eigevalues λ 1 = d λ 2 λ. Let S, T V be two subsets, ad let E(S, T ) = {s S, t T : {s, t} E} be the set of edges betwee the elemets of S ad T. The we have d S T E(S, T ) λd S T. Proof. Let {v 1,..., v } be a eigebasis correspodig to the eigevalues λ 1... λ. As we saw i the proof of the Average Degree Lemma, we ca expad v(s) ad v(t ) ito v(s) = α i v i ad v(t ) = β i v i. For all i [], let λ i be the eigevalue correspodig to eigevector v i. The we have ( ( ) E(S, T ) = v(s) t A v(t ) = α i v i )ta β i v i = λ i α i β i. We kow the graph is d-regular, ad so every vertex has d edges comig out of it. Thus whe we apply A to the vector 1 = (1, 1, 1,...), the result is d times the vector 1. To ormalize this quatity, we divide it by its orm, ad the result we get is v 1. The, sice v 1 = 1 1 = 1, we coclude α 1 = v(s), v 1 = S I [1] it is show that the largest eigevalue λ 1 d. Thus we kow the eigevalue of 1 is d, which proves that λ 1 = d. The we get E(S, T ) = λ 1 α 1 β 1 + λ i α i β i = d i=2 S T + λ i α i β i. max λ i λ>1 Give λ =, we kow λ i λ d, for all i > 1. The we deduce d d S T E(S, T ) λ i α i β i λ d α i β i. By Cauchy-Schwartz Iequality, we ca deduce d S T E(S, T ) λ d v(s) v(t ) = λ d S T. i=2 To grasp Wilf s Theorem, which allows us to determie elusive properties of graphs with the tool of eigevalues, we eed the followig defiitio. Defiitio A legal k-colorig of a graph G is a map c : V {1, 2,..., k} such that for all {i, j} E we have c(i) c(j). The chromatic umber of G, deoted χ(g), is the smallest value of k for which a legal k-colorig exists. I simpler terms, the chromatic umber of a graph is the smallest umber of colors eeded to assig colors to the vertices of a graph without assigig the same color to ay pair of adjacet vertices. Theorem 3.12 (Wilf s Theorem). χ(g) 1 + λ 1. i=2 i=2.

8 8 JINGJING (JENNY) LI Wilf s Theorem reveals how a graph is cotrolled by eigevalues. Determiig colorig is laborious, ad there is o quick ad easy way to do so without goig through ad checkig all possible colorigs. Thus, Wilf s Theorem is very powerful because it guaratees that there exists oe colorig ad a upper boud of the umber of colors eeded to color the graph, without havig to kow ay specific colorig of the graph. Proof. To prove Wilf s Theorem, it is sufficiet to show there exists a orderig for the vertices v 1, v 2,..., v such that for all i [], the umber of eighbors that v i has amog the oes that come before it is at most λ 1. If we ca order the vertices this way, the we ca color the vertices oe by oe with λ may colors. Now we wat to show such a orderig exists. We prove by iductio o i that we ca costruct v i. We begi with i = 0 ad show that we ca choose v with at most λ 1 may eighborig vertices. Let V be the set of all vertices v 1, v 2,..., v. We kow such v exists because by Lemma 3.8, deg ave V λ 1, which implies that there exists v V such that deg V (v ) λ 1. For the iductio hypothesis assume that we have already chose v k+1,..., v. We wat to show that we ca choose v k. Let S be the collectio of all vertices except for v k+1,..., v. The we get a subgraph with k vertices. Applyig Lemma 3.8 to S, we kow there exists v k S such that deg S (v k ) λ 1. By reverse iductio, we have thus prove that we ca costruct a orderig of vertices such that every vertex v i has at most λ 1 may eighbors amog the previous vertices. I coclusio, the Spectral Theorem ca reveal deep isights ito a graph, as demostrated by its applicatios i graph theory such as Wilf s Theorem. This is exactly the beauty of the Spectral Theorem, which is uveilig rather iaccessible aspects of the graph by providig easily computable ivariats of a graph, i the form of eigevalues. Ackowledgmets. It is a pleasure to thak my metor, Nir Gadish, who has dedicated sigificat amout of time ad effort to my research edeavor. I sicerely appreciate his geerous support ad help with formalizig the proof of the Spectral Theorem, ad explorig elighteig cosequeces ad iterestig applicatios i spectral graph theory. Refereces [1] Daiel A. Spielma. Chapter 16 Spectral Graph Theory. spielma/papers/sgtchapter.pdf [2] Daiel A. Spielma. Spectral Graph Theory Lecture 3: The Adjacecy Matrix ad Graph Colorig. [3] Daiel A. Spielma. Spectral Graph Theory Lecture 4: Courat-Fischer ad Graph Colorig. [4] Fa Chug Graham. Chapter 1: Eigevalues ad the Laplacia of a graph. ucsd.edu/~fa/research/cb/ch1.pdf [5] Laszlo Babai. Liear algebra ad applicatios to graphs Part II. edu/~laci/reu01/lialg2.pdf [6] Madhur Tulsiai. REU 2013: Appretice Program Lecture 7: July 8, uchicago.edu/~madhurt/courses/reu2013/lecture7.pdf [7] Madhur Tulsiai. REU 2013: Appretice Program Lecture 15: July 22, uchicago.edu/~madhurt/courses/reu2013/class722.pdf

9 SPECTRAL THEOREM AND APPLICATIONS 9 [8] Salil Vadha. CS225: Pseudoradomess Lecture 7: Expader Graphs. harvard.edu/~salil/cs225/sprig07/lecotes/lec7.pdf [9] Thomas Sauerwald ad He Su. Spectral Graph Theory ad Applicatios Lecture 3: Expader Mixig Lemma. ws11/sgt/lecture3.pdf