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### hw5

Course: ECE 210, Spring 2011
School: UCSB
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2010 Homework ECE210A Fall 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 nd 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, nd all matrices A for...

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2010 Homework ECE210A Fall 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 nd 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, nd all matrices A for which R(X ) = R(A) and R(AT ) = R(Y ). 5. Let X and Y be two full column rank real matrices. What are the conditions (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 satised. How many linearly independent equations were there? What was the dimensionality of the and left right nullspaces? Did the rank-nullity theorem hold? 6. Suppose A Rmn , y Rm , and 0 = s Rn . Show that E = (y As)sT /sT 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 TA = Y, XT . From the set of all such matrices A, nd the one that has the least Frobenius norm. 8. Let M be a real symmetric n n matrix, and let LM be a linear operator from Rnn to Rnn , dened by the equation LM (A) = AT M + M A . 2 Dene the matrix (or operator) 2-norm of LM (denoted by LM LM 2 = max A F =1 LM (A) Show that LM Extra credit: Show that LM 2 2 M =M 1 2. F. F. 2) by
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