电子科技大学：《矩阵理论 Matrix Theory》课程教学资源（课件讲稿）10 matrix norm

Matrix Theory -Matrix Norms School of Mathematical Sciences Teaching Group Textbook: llse C.F.Ipsen,Numerical Matrix Analysis:Linear Systems and Least Squares.SIAM, 2009. Reference books: Fuzhen Zhang.Matrix Theory-Basic Results and Techniques,Second Edition. Springer,2011. Roger A.Horn and Charles A.Johnson:Matrix Analysis.Cambridge University Press,1985. Gene H.Golub and Charles F.Van Loan:Matrix Computations,Third Edition. Johns Hopkins Press,1996. Nicholas J.Higham.Accuracy and Stability of Numerical Algorithms,Second Edition.SIAM,2002. Y.Saad.Iterative Methods for Sparse Linear Systems,Second Edition.SIAM, Philadelphia,2003. Matrix Theory Matrix Norms Maintained by Yan-Fei Jing

Textbook: Ilse C. F. Ipsen, Numerical Matrix Analysis: Linear Systems and Least Squares. SIAM, 2009. Reference books: ▸ Fuzhen Zhang. Matrix Theory-Basic Results and Techniques, Second Edition. Springer, 2011. ▸ Roger A. Horn and Charles A. Johnson: Matrix Analysis. Cambridge University Press, 1985. ▸ Gene H. Golub and Charles F. Van Loan: Matrix Computations, Third Edition. Johns Hopkins Press, 1996. ▸ Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, Second Edition. SIAM, 2002. ▸ Y. Saad. Iterative Methods for Sparse Linear Systems, Second Edition. SIAM, Philadelphia, 2003. Maintained by Yan-Fei Jing Matrix Theory ––Matrix Norms School of Mathematical Sciences Teaching Group Matrix Theory Matrix Norms

Introduction Outline Introduction Definition Induced Norm-Matrix p-Norms One Norm Infinity Norm Norm of a Product Two Norm Frobenius Norm Norm of a Submatrix Exercises Comprehensive Problems 色电有这女子 Matrix Theory Matrix Norms -2/35

Introduction Outline Introduction Definition Induced Norm–Matrix p-Norms One Norm Infinity Norm Norm of a Product Two Norm Frobenius Norm Norm of a Submatrix Exercises Comprehensive Problems Matrix Theory Matrix Norms - 2/35

Introduction Introduction The norm is a useful quantity which can give important information about a matrix. 命电有这女 Matrix Theory Matrix Norms -3/35

Introduction Introduction ▸ The norm is a useful quantity which can give important information about a matrix. ▸ A matrix norm is a number defined in terms of the entries of the matrix. ▸ The norm of a matrix is a measure of how large its elements are. ▸ The analysis of matrix-based algorithms often requires use of matrix norms. ▸ These algorithms need a way to quantify the “size” of a matrix or the “distance” between two matrices that is not necessarily related to how many rows or columns the matrix has. Matrix Theory Matrix Norms - 3/35

Introduction Introduction The norm is a useful quantity which can give important information about a matrix. A matrix norm is a number defined in terms of the entries of the matrix. 奇电有这头 Matrix Theory Matrix Norms -3/35

Introduction Introduction ▸ The norm is a useful quantity which can give important information about a matrix. ▸ A matrix norm is a number defined in terms of the entries of the matrix. ▸ The norm of a matrix is a measure of how large its elements are. ▸ The analysis of matrix-based algorithms often requires use of matrix norms. ▸ These algorithms need a way to quantify the “size” of a matrix or the “distance” between two matrices that is not necessarily related to how many rows or columns the matrix has. Matrix Theory Matrix Norms - 3/35

Introduction Introduction The norm is a useful quantity which can give important information about a matrix. .A matrix norm is a number defined in terms of the entries of the matrix. The norm of a matrix is a measure of how large its elements are. 奇电有这头 Matrix Theory Matrix Norms -3/35

Introduction Introduction ▸ The norm is a useful quantity which can give important information about a matrix. ▸ A matrix norm is a number defined in terms of the entries of the matrix. ▸ The norm of a matrix is a measure of how large its elements are. ▸ The analysis of matrix-based algorithms often requires use of matrix norms. ▸ These algorithms need a way to quantify the “size” of a matrix or the “distance” between two matrices that is not necessarily related to how many rows or columns the matrix has. Matrix Theory Matrix Norms - 3/35

Introduction Introduction The norm is a useful quantity which can give important information about a matrix. .A matrix norm is a number defined in terms of the entries of the matrix. The norm of a matrix is a measure of how large its elements are. The analysis of matrix-based algorithms often requires use of matrix norms. 奇电有这头 Matrix Theory Matrix Norms -3/35

Introduction Introduction ▸ The norm is a useful quantity which can give important information about a matrix. ▸ A matrix norm is a number defined in terms of the entries of the matrix. ▸ The norm of a matrix is a measure of how large its elements are. ▸ The analysis of matrix-based algorithms often requires use of matrix norms. ▸ These algorithms need a way to quantify the “size” of a matrix or the “distance” between two matrices that is not necessarily related to how many rows or columns the matrix has. Matrix Theory Matrix Norms - 3/35

Introduction Introduction The norm is a useful quantity which can give important information about a matrix. .A matrix norm is a number defined in terms of the entries of the matrix. The norm of a matrix is a measure of how large its elements are. The analysis of matrix-based algorithms often requires use of matrix norms. These algorithms need a way to quantify the "size"of a matrix or the "distance"between two matrices that is not necessarily related to how many rows or columns the matrix has. 命电有这女子 Matrix Theory Matrix Norms -3/35

Introduction Introduction ▸ The norm is a useful quantity which can give important information about a matrix. ▸ A matrix norm is a number defined in terms of the entries of the matrix. ▸ The norm of a matrix is a measure of how large its elements are. ▸ The analysis of matrix-based algorithms often requires use of matrix norms. ▸ These algorithms need a way to quantify the “size” of a matrix or the “distance” between two matrices that is not necessarily related to how many rows or columns the matrix has. Matrix Theory Matrix Norms - 3/35

Introduction Conditioning of matrix norms a nonsingular linear system a perturbed system. 命电有这女子 Matrix Theory Matrix Norms -4/35

Introduction Conditioning of matrix norms { Ax = b, a nonsingular linear system; Ax˜ = b˜, a perturbed system. Matrix Theory Matrix Norms - 4/35

Introduction Conditioning of matrix norms Ax b,a nonsingular linear system; Ax=B,a perturbed system. The normwise absolute error is Ix-x=IA-1(b-b)川 命电有这女 Matrix Theory Matrix Norms -4/35

Introduction Conditioning of matrix norms { Ax = b, a nonsingular linear system; Ax˜ = b˜, a perturbed system. The normwise absolute error is ∥x − x˜∥ = ∥A −1 (b − b˜)∥ Matrix Theory Matrix Norms - 4/35

Introduction Conditioning of matrix norms Ax=b,a nonsingular linear system; Ax=B,a perturbed system. The normwise absolute error is x-=A-(b-B)l In order to isolate the perturbation and derive a bound of the form IA--116-bl, we have to define a norm for matrices. 争老年这大习 Matrix Theory Matrix Norms -4/35

Introduction Conditioning of matrix norms { Ax = b, a nonsingular linear system; Ax˜ = b˜, a perturbed system. The normwise absolute error is ∥x − x˜∥ = ∥A −1 (b − b˜)∥ In order to isolate the perturbation and derive a bound of the form ∥A −1 ∥∥b − b˜∥, we have to define a norm for matrices. Matrix Theory Matrix Norms - 4/35