Discrete Math Class 19 ( )

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1 Discrete Math Class 19 ( ) Instructor: László Babai Notes taken by Jacob Burroughs Partially revised by instructor Review: Tuesday, December 6, 3:30-5:20 pm; Ry-276 Final: Thursday, December 8, 10:30-12:30; Ry-251 DO IMPORTANT. Study the relevant chapters of LN (about Markov Chains) and the Linear Algebra online text Similar matrices Definition Let A, B M n (F) where F is a field (i.e. R, C) A, B are similar if C M n (F), C 1 such that B = C 1 AC. Notation A B. DO Similarity is an equivalence relation on M n (F) DO If A B then trace(a) = trace(b). (Hint trace(cd) = trace(dc)) DO If A B then det(a) = det(b). (Hint det(cd) = det(c) det(d)) DO If A B then f A = f B (Their characteristic polynomials are equal) 19.2 Matrix of a linear map Definition A linear transformation is a function f : V V where f(a + b) = f(a) + f(b) and f(λa) = λf(a). Equivalently, f( α i a i ) = α i f(a i ) Definition A linear map is a function f : V W with the same attributes. DO If v 1,..., v n are a basis of V, w 1,..., w n are arbitrary vectors in W, there exists a unique linear map such that ( i)(f(v i ) = w i ) Definition Coordinates: Let e = (e 1,..., e n ) be a basis of V. Then every v V can be uniquely written as v = α i e i. THe α i are the coordinates of v wrt (with respect to) the basis e.arranged in a column vector, we write [v] e = (alpha 1,..., α n ) T (transpose, to make it a column vector) Change of basis Definition (Change of basis matrix). Take two bases: e = (e 1,..., e n ) (the old basis) and e = (e 1,..., e n). The change of basis matrix is S = [[e 1] e,..., [e n] e ]. (The i-th column lists the coordinates of e i wrt e.) 1

2 DO (Change of coordinates under change of basis). [v] new = S 1 [v] old DO [e] e = [e ] 1 e Definition (Matrix of a linear map). Let ϕ : V W be a linear map. Let e = (e 1,..., e n ) be a basis of V and f = (f 1,..., f k ) be a basis of W. The matrix of ϕ wrt this pair of bases is [ϕ] e,f = [[ϕ(e 1 ) f,..., ϕ(e n ) f ] So this is a k n matrix. DO (Change of matrix under change of bases). Let us have a linear map ϕ : V W. Let e, f be old bases of V and W respectively, and e, f be new bases. Then define S, T as the change of basis matrices. Let A = [ϕ] e,f and A = [ϕ] e,f Then A = T 1 AS Corollary If ϕ : V V, then [ϕ] new = S 1 [ϕ] old S Corollary A B if and only if ϕ : V V and bases e, e such that A = [ϕ] e and B = [ϕ] e Corollary A linear transformation has a characteristic polynomial. (because similar matrices have the same characteristic polynomial) λ Definition A is diagonalizable if A a diagonal matrix = D = λ n Then f A (t) = f D (t) = (t λ i ) [ ] 1 1 Example A = is not diagonalizable (since it is not similar to I). 0 1 DO A matrix is diagonalizable if and only if it has an eigenbasis Hint: A = [ϕ] e and make e the eigenbasis. Make S = [e ] e Then S 1 AS is a diagonal matrix D. You need to show that AS = SD Corollary If f A has n distinct roots, then A is diagonalizable. Caveat: I is diagonalizable, yet it has multiple eigenvalues. DO A B = rk A = rk B DO ( A)( S)(if S nonsingular then rk(as) = rk A) DO rk(ab) rk(a) and rk(ab) rk(b) 2

3 19.4 Eigensubspaces. Geometric and algebraic multiplicity of eigenvalues Definition Algebraic multiplicity of eigenvalue λ is its multiplicity in the characteristic polynomial, i.e., it is the largest m such that (t λ) m f A. Definition Geometric multiplicity: The maximum number of linearly independent eigenvectors to λ: U λ = {x Ax = λx} F n This is the eigensubspace to λ. DO λ is an eigenvalue if and only if dim U λ 1 The geometric multiplicity of λ is the dimension of U λ DO dim U λ = n rk(λi A) Hint: Rank nullity DO The geometric multiplicity of λ is less than or equal to the algebraic multiplicity of λ. DO Over C: λ C algebraic multiplicity of λ = n. DO Over C: A is diagonalize if and only if λ C geometric multiplicity of λ = n λ, the algebraic and geometric multiplicities are equal. DO * Over C, every matrix is similar to a triangular matrix (This is a hint for the above exercise) 19.5 Norm, orthogonality in R n Over R: The standard dot product in R n : x y = x T y = x i y i x and y are orthogonal if x y = 0 We define the norm x = x x = x 2 i DO Cauchy-Schwarz inequality: a b a b DO Triangle inequality: a + b a + b DO Show Cauchy-Schwarz is equivalent to the triangle inequality. DO If v 1,..., v k R n are orthogonal and non-zero, they are linearly independent. Definition The operator norm of A R k n : A = sup Ax x 3

4 DO Show this supremum is a maximum. DO If A = (α ij ) then A α ij Theorem (Spectral theorem). If A M n (R) and A = A T (A is a symmetric real matrix)then A has an orthonormal{ eigenbasis, i.e., ( b 1,... b n R n 1 i = j )(b i b j = δ ij = 0 i j, b i = 1, and Ab i = λ i b i ) Any orthonormal system of eigenvectors can be extended to an orthonormal eigenbasis. DO If A = A T (A is symmetric) then A = λ max = max i Λ i. Hint: Spectral theorem 19.6 Elements of spectral graph theory DO A connected undirected graph is aperiodic if and only if it is not bipartite. { 1 i j Definition The adjacency matrix of a graph: A g = (a ij ) where a ij = 0 i j DO The graph G is regular of degree r if and only if 1 is an eigenvector to eigenvalue r. DO For an r-regular graph G, let λ 1,..., λ n be the eigenvalues of the adjacency matrix. (Note: these are real because the adjacency matrix is symmetric.) λ, λ r DO Let G be a regular graph of degree r with eigenvalues r = λ 1 λ 2 λ n. Prove: λ 2 = r if and only if G is disconnected. (a) Show that this follows from Perron Frobenius. (b) Prove this without using Perron Frobenius. DO Let G be a connected regular graph of degree r. Then r is an eigenvalue if and only if G is bipartite Rate of convergence of random walk on a graph: spectral estimate Definition (SLEM). Let A be a symmetric real matrix with eigenvalues λ 1 λ 2 λ n. Then SLEM(A) = max i λ i = max λ 2, λ n (second largest eigenvalue modulus) Notation. J is the all-ones matrix. Theorem (Convergence rate of naive random walk on regular graph). Let G be a connected regular non-bipartite graph. Let A be the adjacency matrix of A. So T = (1/r)A is the transition matrix of the naive random walk on G. Then lim T t = 1 J and T t 1 J λt n n where λ is SLEM(T ) = (1/r)SLEM(A) (so 0 < λ < 1). 4

5 Proof. We shall show that the maximum absolute value of the eigenvalues of T t 1 n J is λt. The all-ones vector 1 is an eigenvector of T to eigenvalue 1. Let e 1 be its normalized value: e 1 = (1/ n)1. Let e 1, e 2,..., e n be an orthonormal eigenbasis of T. We claim that e 1, e 2,..., e n is also an orthonormal eigenbasis of T t (1/n)J. For i 2 we have e i e 1 and therefore e i 1. Therefore, for i 2 we have Je i = 0 and therefore (T t (1/n)J)e i = T t e i = λ t ie i. Also, T t e 1 = e 1 (because T is stochastic and therefore T t is stochastic), and the same holds for (1/n)J, so (1/n)Je 1 = e 1. Therefore (T t (1/n)J)e 1 = 0. So e 1, e 2,..., e n form an orthonormal eigenbasis of T t (1/n)J with eigenvalues (in this order) 0, λ t 2,..., λ t n. The maximum absolute value among these is therefore λ t, as claimed. 5