Definition Conditioning of matrix norms Matrix norms are well-conditioned,in the absolute sense and in the relative sense. That is,if A,Ee Cmxn,then IIA+EI-IAI≤IEI 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∥. 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,Ee Cmxn,then IIA+EI-IAI≤IEI Proof. 奇电有头子 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. 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,Ee Cmxn,then IIA+EI-IAI≤IEI: Proof. A+EA+E,from the triangle inequality: 奇电有这头子 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; 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,Ee Cmx,then IIA+EI-IAII≤IEI Proof. A+ElA+E,from the triangle inequality: 了IA+EI-IAl≤IEI, 奇电有这头 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 + E∥ − ∥A∥ ≤ ∥E∥, 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,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 了IA+EI-IA≤IEI 命电有这女 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∥, Matrix Theory Matrix Norms - 7/35