Science:Math Exam Resources/Courses/MATH307/December 2012/Question 06 (c)
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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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Hint 

Use the fact that U and V are unitary and hence do not affect the norm of a vector. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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 . 