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MNorms - Norms of Matrices We can measure matrix sizes...

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Norms of Matrices We can measure matrix sizes using vector norms, because R m × n is a vector space. Although the names are different, all of the p-norms above give matrix norms if the matrix is stretched into one long vector. The only matrix norm of this flavor that is used often is the Frobenius norm k A k F = ( m X i =1 n X j =1 | a ij | 2 ) 1 / 2 = tr( A t A ) 1 / 2 . It is the vector 2-norm applied to a stretched out version of the matrix. Any norm on the vector space R mn can be used as a matrix norm on R m × n . But matrices are also operators, and as such we can measure their size by how much they stretch the vectors on which they operate: k A k D , R = max k x k D =1 k Ax k R . You can think of it as the radius of the smallest ball centered at the origin of R m that contains the image of the ball of radius 1 centered at the origin of R n (the unit ball in R n ). This is the norm induced by (or subordinate to) the vector norms k · k D and k · k R . For some authors, this is the only type of matrix norm.
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