AMS526: Numerical Analysis I (Numerical Linear Algebra)
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1 AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Matrix Norms; Singular Value Decomposition Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 13
2 Outline 1 Matrix Norms 2 Singular Value Decomposition Xiangmin Jiao Numerical Analysis I 2 / 13
3 Matrix Norms Induced by Vector Norms Viewing m n matrix as mn-vectors is not always useful, as operations involving m n matrices do not behave this way Induced matrix norms capture such behavior Xiangmin Jiao Numerical Analysis I 3 / 13
4 Matrix Norms Induced by Vector Norms Viewing m n matrix as mn-vectors is not always useful, as operations involving m n matrices do not behave this way Induced matrix norms capture such behavior Definition Given vector norms (n) and (m) on domain and range of A C m n, respectively, the induced matrix norm A (m,n) is the smallest number C R for which the following inequality holds for all x C n : Ax (m) C x (n). Xiangmin Jiao Numerical Analysis I 3 / 13
5 Matrix Norms Induced by Vector Norms Viewing m n matrix as mn-vectors is not always useful, as operations involving m n matrices do not behave this way Induced matrix norms capture such behavior Definition Given vector norms (n) and (m) on domain and range of A C m n, respectively, the induced matrix norm A (m,n) is the smallest number C R for which the following inequality holds for all x C n : Ax (m) C x (n). In other words, it is supremum of ratio Ax (m) / x (n) for all nonzero vectors x C n Maximum factor by which A can stretch x C n A (m,n) = sup Ax (m) / x (n) = sup Ax (m). x C n,x 0 x C n, x (n) =1 Xiangmin Jiao Numerical Analysis I 3 / 13
6 Matrix Norms Induced by Vector Norms Viewing m n matrix as mn-vectors is not always useful, as operations involving m n matrices do not behave this way Induced matrix norms capture such behavior Definition Given vector norms (n) and (m) on domain and range of A C m n, respectively, the induced matrix norm A (m,n) is the smallest number C R for which the following inequality holds for all x C n : Ax (m) C x (n). In other words, it is supremum of ratio Ax (m) / x (n) for all nonzero vectors x C n Maximum factor by which A can stretch x C n A (m,n) = sup Ax (m) / x (n) = sup Ax (m). x C n,x 0 x C n, x (n) =1 Is vector norm consistent with matrix norm of m 1-matrix? Xiangmin Jiao Numerical Analysis I 3 / 13
7 1-norm By definition A 1 = sup Ax 1 x C n, x 1 =1 Xiangmin Jiao Numerical Analysis I 4 / 13
8 1-norm By definition A 1 = sup Ax 1 x C n, x 1 =1 What is it equal to? Xiangmin Jiao Numerical Analysis I 4 / 13
9 1-norm By definition A 1 = sup Ax 1 x C n, x 1 =1 What is it equal to? Maximum of 1-norm of column vectors of A maximum column sum of A is oversimplified in the textbook Xiangmin Jiao Numerical Analysis I 4 / 13
10 1-norm By definition A 1 = sup Ax 1 x C n, x 1 =1 What is it equal to? Maximum of 1-norm of column vectors of A maximum column sum of A is oversimplified in the textbook To show it, note that for x C n and x 1 = 1 n Ax 1 = x j a j 1 j=1 n j=1 x j a j 1 max 1 j n a j 1 x 1 Let k = arg max 1 j n a j 1, then Ae k 1 = a k 1, so max 1 j n a j 1 is tight upper bound Xiangmin Jiao Numerical Analysis I 4 / 13
11 -norm By definition A = What is A equal to? sup Ax x C n, x =1 Xiangmin Jiao Numerical Analysis I 5 / 13
12 -norm By definition A = What is A equal to? sup Ax x C n, x =1 Maximum of 1-norm of column vectors of A To show it, note that for x C n and x = 1 Ax = max 1 i m a i x max 1 i m a i 1 x where a i denotes ith row vector of A Furthermore, max 1 i m a i 1 is a tight bound. Which vector can we choose to reach the bound? Xiangmin Jiao Numerical Analysis I 5 / 13
13 2-norm What is 2-norm of a matrix? Xiangmin Jiao Numerical Analysis I 6 / 13
14 2-norm What is 2-norm of a matrix? Answer: Its largest singular value. We will talk more about singular-value decomposition Xiangmin Jiao Numerical Analysis I 6 / 13
15 2-norm What is 2-norm of a matrix? Answer: Its largest singular value. We will talk more about singular-value decomposition What is 2-norm of a diagonal matrix? Xiangmin Jiao Numerical Analysis I 6 / 13
16 Cauchy-Schwarz and Hölder Inequalities Hölder inequality: Let p and q satisfy 1/p + 1/q = 1 with 1 p, q, then x y x p y q Cauchy-Schwarz inequality x y x 2 y 2 Cauchy-Schwarz inequality is a special case of Hölder inequality Xiangmin Jiao Numerical Analysis I 7 / 13
17 Cauchy-Schwarz and Hölder Inequalities Hölder inequality: Let p and q satisfy 1/p + 1/q = 1 with 1 p, q, then x y x p y q Cauchy-Schwarz inequality x y x 2 y 2 Cauchy-Schwarz inequality is a special case of Hölder inequality Example: What is 2-norm of rank-one matrix? Hint: Use Cauchy-Schwarz inequality. Xiangmin Jiao Numerical Analysis I 7 / 13
18 Bounding Matrix-Matrix Multiplication Let A be an l m matrix and B an m n matrix, then for x C n AB (l,n) A (l,m) B (m,n) Xiangmin Jiao Numerical Analysis I 8 / 13
19 Bounding Matrix-Matrix Multiplication Let A be an l m matrix and B an m n matrix, then for x C n AB (l,n) A (l,m) B (m,n) To show it, note ABx (l) A (l,m) Bx (m) A (l,m) B (m,n) x (n), Xiangmin Jiao Numerical Analysis I 8 / 13
20 Bounding Matrix-Matrix Multiplication Let A be an l m matrix and B an m n matrix, then for x C n AB (l,n) A (l,m) B (m,n) To show it, note ABx (l) A (l,m) Bx (m) A (l,m) B (m,n) x (n), In general, this inequality is not an equality In particular, A n A n but A n A n in general for n 2 Xiangmin Jiao Numerical Analysis I 8 / 13
21 General Matrix Norms One can view m n matrices as mn-dimensional vectors and obtain general matrix norms, which satisfy (for A, B C m n ) (1) A 0, and A = 0 only if A = 0, (2) A + B A + B, (3) αa = α A. Xiangmin Jiao Numerical Analysis I 9 / 13
22 Frobenius Norm One useful norm is Frobenius norm (a.k.a. Hilbert-Schmidt norm) m n n A F = a ij 2 = a j 2 2 i.e., 2-norm of mn-vector Furthermore, i=1 j=1 A F = tr(a A) where tr(b) denotes trace of B, the sum of its diagonal entries j=1 Xiangmin Jiao Numerical Analysis I 10 / 13
23 Frobenius Norm One useful norm is Frobenius norm (a.k.a. Hilbert-Schmidt norm) m n n A F = a ij 2 = a j 2 2 i.e., 2-norm of mn-vector Furthermore, i=1 j=1 A F = tr(a A) where tr(b) denotes trace of B, the sum of its diagonal entries j=1 Note that because AB 2 F = AB F A F B F n m n m a i b j 2 ( a i 2 b j 2 ) 2 = A 2 F B 2 F i=1 j=1 i=1 j=1 Xiangmin Jiao Numerical Analysis I 10 / 13
24 Invariance under Unitary Multiplication Theorem For any A C m n and unitary Q C m m, we have QA 2 = A 2 and QA F = A F. In other words, 2-norm and Frobenius norms are invariant under unitary multiplication. Xiangmin Jiao Numerical Analysis I 11 / 13
25 Invariance under Unitary Multiplication Theorem For any A C m n and unitary Q C m m, we have QA 2 = A 2 and QA F = A F. In other words, 2-norm and Frobenius norms are invariant under unitary multiplication. Proof for 2-norm: Qy 2 = y 2 for y C m and therefore QAx 2 = Ax 2 for x C n. It then follows from definition of 2-norm. Xiangmin Jiao Numerical Analysis I 11 / 13
26 Invariance under Unitary Multiplication Theorem For any A C m n and unitary Q C m m, we have QA 2 = A 2 and QA F = A F. In other words, 2-norm and Frobenius norms are invariant under unitary multiplication. Proof for 2-norm: Qy 2 = y 2 for y C m and therefore QAx 2 = Ax 2 for x C n. It then follows from definition of 2-norm. Proof for Frobenius norm: QA 2 F = tr ((QA) QA) = tr (A Q QA) = tr (A A) = A 2 F. Xiangmin Jiao Numerical Analysis I 11 / 13
27 Outline 1 Matrix Norms 2 Singular Value Decomposition Xiangmin Jiao Numerical Analysis I 12 / 13
28 Geometric Observation The image of unit sphere under any m n matrix is a hyperellipse Give a unit sphere S in R n, let AS denote the shape after transformation SVD is A = UΣV where U C m m and V C n n is unitary and Σ R m n is diagonal Singular values are diagonal entries of Σ, correspond to the principal semiaxes, with entries σ 1 σ 2 σ n 0. Left singular vectors of A are column vectors of U and are oriented in the directions of the principal semiaxes of AS Right singular vectors of A are column vectors of V and are the preimages of the principal semiaxes of AS Av j = σ j u j for 1 j n Xiangmin Jiao Numerical Analysis I 13 / 13
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