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