Definition 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 -5/35
Definition 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 - 5/35
Definition Definition A matrix norm is a function from Cmxn to R with three properties: 命电有这女子 Matrix Theory Matrix Norms -6/35
Definition Definition A matrix norm ∣∣ ⋅ ∣∣ is a function from C m×n to R with three properties: Matrix Theory Matrix Norms - 6/35
Definition Definition A matrix norm‖·‖is a function from cmxn to R with three properties: Nonnegative:‖A≥0 for all A∈Cmxn, All =0 if and only if A=0. 奇电有这头 Matrix Theory Matrix Norms -6/35
Definition Definition A matrix norm ∣∣ ⋅ ∣∣ is a function from C m×n to R with three properties: Nonnegative: ∣∣A∣∣ ≥ 0 for all A ∈ C m×n , ∣∣A∣∣ = 0 if and only if A = 0. Homogeneous: ∣∣αA∣∣ = ∣α∣∣∣A∣∣ for all α ∈ C, A ∈ C m×n . Triangle inequality: ∣∣A + B∣∣ ≤ ∣∣A∣∣ + ∣∣B∣∣ for all A,B ∈ C m×n . Matrix Theory Matrix Norms - 6/35
Definition Definition A matrix normll is a function from Cmxn to R with three properties: Nonnegative:l‖Al≥0 for all ACmxn, Al =0 if and only if A=0. Homogeneous:llaAll lallAll for all a C.ACmxn. 奇电有头子 Matrix Theory Matrix Norms -6/35
Definition Definition A matrix norm ∣∣ ⋅ ∣∣ is a function from C m×n to R with three properties: Nonnegative: ∣∣A∣∣ ≥ 0 for all A ∈ C m×n , ∣∣A∣∣ = 0 if and only if A = 0. Homogeneous: ∣∣αA∣∣ = ∣α∣∣∣A∣∣ for all α ∈ C, A ∈ C m×n . Triangle inequality: ∣∣A + B∣∣ ≤ ∣∣A∣∣ + ∣∣B∣∣ for all A,B ∈ C m×n . Matrix Theory Matrix Norms - 6/35
Definition Definition A matrix normll is a function from Cmxn to R with three properties: Nonnegative:‖All≥0 for all A∈Cmxn, Al =0 if and only if A=0. Homogeneous:llaAll lalllAll for all a eC.Ae cmxn. Triangle inequality:‖A+Bll≤A+BI for all A,B∈Cmxn 奇电有头子 Matrix Theory Matrix Norms -6/35
Definition Definition A matrix norm ∣∣ ⋅ ∣∣ is a function from C m×n to R with three properties: Nonnegative: ∣∣A∣∣ ≥ 0 for all A ∈ C m×n , ∣∣A∣∣ = 0 if and only if A = 0. Homogeneous: ∣∣αA∣∣ = ∣α∣∣∣A∣∣ for all α ∈ C, A ∈ C m×n . Triangle inequality: ∣∣A + B∣∣ ≤ ∣∣A∣∣ + ∣∣B∣∣ for all A,B ∈ C m×n . Matrix Theory Matrix Norms - 6/35