MAT 4725 Numerical Analysis Section 7.1 Part I Norms of Vectors and Matrices

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Presentation transcript:

MAT 4725 Numerical Analysis Section 7.1 Part I Norms of Vectors and Matrices

Chapter 7 Iterative Techniques in Matrix Algebra Section 2.2 Fixed-Point Iteration

Chapter 7 Iterative Techniques in Matrix Algebra Section 7.3 Iterative Techniques

Chapter 7 Iterative Techniques in Matrix Algebra Section 7.3 Iterative Techniques

Chapter 7 Iterative Techniques in Matrix Algebra Section 7.3 Iterative Techniques

Chapter 7 Iterative Techniques in Matrix Algebra Section 7.3 Iterative Techniques We need a way to measure the distance between two vectors

7.1 Norms of Vectors and Matrices Norms on real vector space (Part I) Norms on Matrices (Part II)

n Dimensional Real Vector Space Linear Algebra, Mult. Variables Calculus Applied Analysis

Definition 7.1

l 2 Norm (Euclidean Norm) Usual distance function on

l 2 Norm (Euclidean Norm) Usual distance function on Geometric Interpretations of

l  Norm Geometric Interpretations of

Example 1

Theorem 7.3 (Cauchy-Schwarz Inequality)

l 2 Norm is a vector norm Proof:

Definition 7.4

Definition 7.5

Theorem

Example 2

Classwork

Homework Download HW. Read 7.1 (skip all the proofs)