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

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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 ...
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