midterm - 18.335 Midterm You have two hours All problems...

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18.335 Midterm You have two hours. All problems have equal weight , but some problems may be harder than others, so try not to spend too much time on one problem. Problem 1: Schur, backsubstitution, complexity (20 points) You are given matrices A ( m × m ), B ( n × n ), and C ( m × n ), and want to solve for an unknown matrix X ( m × n ) solving: AX - XB = C . We will do this using the Schur decompositions of A and B . (Recall that any square matrix S can be factorized in the Schur form S = QUQ * for some unitary matrix Q and some upper-triangular matrix U .) (a) Given the Schur decompositions A = Q A U A Q * A and B = Q B U B Q * B , show how to transform AX - XB = C into new equations A 0 X 0 - X 0 B 0 = C 0 , where A 0 and B 0 are upper triangular and X 0 is the new m × n matrix of unknowns. That is, express, A 0 , B 0 , and C 0 in terms of A , B , and C (or their Schur factors), and show how to get X back from X 0 . (b) Given the upper-triangular system A 0 X 0 - X 0 B 0 = C 0 from (a), give an algorithm to find the last row of X 0 . (Hint: look at the last row of the equation.) (c) Suppose you have computed all rows > j of X 0 , give an algorithm to compute the j -th row. (d) The combination of the previous three parts yields a backsubstitution-like algorithm to solve for
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midterm - 18.335 Midterm You have two hours All problems...

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