HW3 Sol - EE 103 Winter 09 Prof SEJ HW 3 Sol Applied...

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EE 103, Winter ’09, Prof SEJ: HW 3 Sol. Page 1 / 12 Applied Numerical Computing Instructor: Prof. S. E. Jacobsen HW3 Solution Students: Distributed HW solutions are a component of the course and should be fully understood. Problem 1) We have included “arrows” to remind you that the components, especially the “ i b ”, are vectors. We know that we can add, transpose, and multiply block matrices if the corresponding blocks have the correct sizes (and if we pay attention to the order of multiplication as we multiply the blocks). We write A and B as follows: 1 1 2 2 ... , and n m n n k n b b A a a a B b × × = = G G G G G # G , where i a G are 1 m × column vectors and j b G are 1 k × row vectors. A and B have the form of two block matrices now and we can treat each block as if it were a number, and as long as the dimensions match, we can proceed to multiply and sum. Normally we have: [ ] 1 2 1 2 1 1 n n n n y y x x x x y x y y = + + " " # so, in the case of block matrices A and B, we can write: 1 2 1 2 2 1 2 1 ... . n n n j n j j n b b AB a a a a b a b a b a b b = = = + + + = G G G G G G G G G G G G G " # G the dimensions of each pair of matrices (row and column vectors) work for multiplications and summation. Each i j a b G G produces an m k × matrix and the result of the sum will also be m k × as we expect. Alternatively, the results can be shown as follows:
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