Chapter11.2 - The norm of a vector is a generalization of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The norm of a vector is a generalization of the concept of its length. The default norm is the Euclidean norm which corresponds to our notion of length >> a=[3 4]; >> norm(a) ans = 5 However, instead of the square root of the sum of the squares, it is possible to generalize a p = aip 1/ p >> norm(a,1) is the sum of the absolute values of the compoents ans = 7 >> norm(a,3) is the cubic roots of the sums of the cubes ans = 4.4979 >> norm(a,20) ans = 4.0006 >> norm(a,inf) is the absolute values of the largest components. ans = 4 We can calculate the norm of a matrix in a similar way, which is called the Frobenius norm c = 2 1 2 0 >> norm(c,'fro') ans = 3 However, for matrices, the most commonly used norms are the "induced" norms that tell us the largest increase in length that a matrix can effect when it multiplies a vector From Wikipedia which is simply the maximum absolute column sum of the matrix which is simply the maximum absolute row sum of the matrix >> norm(c,1) ans = 4 1 Which we will get if we multiply c by the vector 0 >> norm(c,2) ans = 2.9208 >> norm(c,inf) ans = 3 Which we will get if C multiplies the vector 1 1 The condition number of a matrix is the norm of the matrix times the norm of its inverse. It gives you the maximum amplification of errors in the matrix on the solution >> cinv=inv(c) cinv = 0 0.5000 1.0000 1.0000 >> norm(c) ans = 2.9208 >> norm(cinv) ans = 1.4604 >> cond(c) ans = 4.2656 ...
View Full Document

This note was uploaded on 03/27/2012 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

Ask a homework question - tutors are online