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# hw5 - s T/s T s has the smallest 2-norm of all m × n...

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ECE210A Fall 2010 Homework 4 Due date : November 1, 2010. 1. Reading assigment . Read chapter 4 of the class notes. 2. If X and Y are full column rank matrices of rank p , show that the rank of XY T is equal to p . 3. Show that if A is a rank p matrix then you can find two full column rank matrices X and Y of rank p such that A = XY T . 4. If X and Y are two full column rank matrices of rank p , find all matrices A for which R ( X ) = R ( A ) and R ( A T ) = R ( Y ). 5. Let X and Y be two full column rank real matrices. What are the condi- tions (if any) on X and Y such that there exists a real matrix A such that AX = Y and Y T A = X T ? Find all such A when the conditions are sat- isfied. How many linearly independent equations were there? What was the dimensionality of the left and right nullspaces? Did the rank-nullity theorem hold? 6. Suppose A R m × n , y R m , and 0 6 = s R n . Show that E = ( y - As
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Unformatted text preview: s T /s T s has the smallest 2-norm of all m × n matrices E that satisfy ( A + E ) s = y . 7. Let X and Y be two full column rank matrices. We know from problem 5 that there are many matrices A which satisfy the pair of equations AX = Y, Y T A = X T . From the set of all such matrices A , ﬁnd the one that has the least Frobe-nius norm. 8. Let M be a real symmetric n × n matrix, and let L M be a linear operator from R n × n to R n × n , deﬁned by the equation L M ( A ) = A T M + MA 2 . Deﬁne the matrix (or operator) 2-norm of L M (denoted by k L M k 2 ) by k L M k 2 = max k A k F =1 k L M ( A ) k F . Show that k L M k 2 ≤ k M k F . Extra credit: Show that k L M k 2 = k M k 2 . 1...
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