Norms of MatricesWe can measure matrix sizes using vector norms, becauseRm×nis a vector space.Although the names are different, all of the p-norms above give matrix norms if thematrix is stretched into one long vector. The only matrix norm of this flavor that isused often is the Frobenius normkAkF= (mXi=1nXj=1|aij|2)1/2= tr(AtA)1/2.It is the vector 2-norm applied to a stretched out version of the matrix. Any normon the vector spaceRmncan be used as a matrix norm onRm×n.But matrices are also operators, and as such we can measure their size by how muchthey stretch the vectors on which they operate:kAkD,R= maxkxkD=1kAxkR.You can think of it as the radius of the smallest ball centered at the origin ofRmthat contains the image of the ball of radius 1 centered at the origin ofRn(theunitballinRn). This is the norminducedby (or subordinate to) the vector normsk · kDandk · kR. For some authors, this is theonlytype of matrix norm.
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