Definition Conditioning of matrix norms Matrix norms are well-conditioned,in the absolute sense and in the relative sense. That is,if A,Ee Cmx,then IIA+EI-IAII≤IEI Proof. IA+EI≤IA+IE from the triangle inequality; A =(A+E)-E<A+El+E,the same reasoning IA+EI-IA≤IEI, -IEI≤I(A+E)-EI-IA 命电有这女子 Matrix Theory Matrix Norms -7/35
Definition Conditioning of matrix norms Matrix norms are well-conditioned, in the absolute sense and in the relative sense. That is, if A,E ∈ C m×n , then ∣ ∥A + E∥ − ∥A∥ ∣ ≤ ∥E∥. Proof. { ∥A + E∥ ≤ ∥A∥ + ∥E∥, from the triangle inequality; ∥A∥ = ∥(A + E) − E∥ ≤ ∥A + E∥ + ∥E∥, the same reasoning. { ∥A + E∥ − ∥A∥ ≤ ∥E∥, − ∥E∥ ≤ ∥(A + E) − E∥ − ∥A∥. Matrix Theory Matrix Norms - 7/35
Induced Norm-Matrix p-Norms 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 -8/35
Induced Norm–Matrix p-Norms 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 - 8/35
nduced Norm-Matrix p-Norms Induced norm-Matrix p-norms LetA∈Cmxn 奇电有这头 Matrix Theory Matrix Norms -9/35
Induced Norm–Matrix p-Norms Induced norm–Matrix p-norms Let A ∈ C m×n . Matrix Theory Matrix Norms - 9/35
Induced Norm-Matrix p-Norms Induced norm-Matrix p-norms Let ACmxn The p-norm Alle max Axlp p21, x+0 Ixlp is a matrix norm. 奇电有这头 Matrix Theory Matrix Norms -9/35
Induced Norm–Matrix p-Norms Induced norm–Matrix p-norms Let A ∈ C m×n . The p-norm ∥A∥p = max x≠0 ∥Ax∥p ∥x∥p , p ≥ 1, is a matrix norm. Matrix Theory Matrix Norms - 9/35
Induced Norm-Matrix p-Norms Induced norm-Matrix p-norms Let ACmxn. The p-norm IlAlle max IAxlp Ixllp p21, ×*0 is a matrix norm. Equivalently,the p-norm of a matrix can be rewritten as IAp=max Axllp:p≥1， xlp=1 奇电有这头 Matrix Theory Matrix Norms -9/35
Induced Norm–Matrix p-Norms Induced norm–Matrix p-norms Let A ∈ C m×n . The p-norm ∥A∥p = max x≠0 ∥Ax∥p ∥x∥p , p ≥ 1, is a matrix norm. Equivalently, the p-norm of a matrix can be rewritten as ∥A∥p = max ∥x∥p=1 ∥Ax∥p, p ≥ 1, Matrix Theory Matrix Norms - 9/35