MATH307 December 2012
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Question 06 (c)
Suppose that A is a matrix with singular value decomposition where
(c) What is the matrix norm ? Give a reason.
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Use the fact that U and V are unitary and hence do not affect the norm of a vector.
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To begin with, recall that unitary matrices don't change the norm of a vector, in particular and for all vectors x. Using the definition of the matrix norm of A we find that
where satisfies . In other words, the matrices and have the same norm. The norm of the diagonal matrix is simply the largest absolute value of the its diagonal entries. Therefore .
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