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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 “ib”, are vectors. We know that we can add, transpose, and multiply block matricesif 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: 1122..., and nm nn knbbAaaaBb××⎡⎤⎢⎥⎢⎥⎡⎤==⎣⎦⎢⎥⎢⎥⎢⎥⎣⎦GGGGG#G, where iaGare 1m×column vectors and jbGare 1k×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: 121211nnnnyyxxxx yx yy⎡⎤⎢⎥⎢⎥=++⎢⎥⎢⎥⎣⎦""#so, in the case of block matrices A and B, we can write: 12122121....nnnjnjjnbbABaaaa ba ba ba bb=⎡⎤⎢⎥⎢⎥⎡⎤⇒==+++=⎣⎦⎢⎥⎢⎥⎢⎥⎣⎦∑GGGGGGGGGGGGG"#Gthe dimensions of each pair of matrices (row and column vectors) work for multiplications and summation. Each ija bGGproduces an mk×matrix and the result of the sum will also be mk×as we expect. Alternatively, the results can be shown as follows:
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