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Unformatted text preview: ECSE 462 Lecture 19 Revisions of Linear Algebra Consider an N-tuple vector = N N x x x x x 1 2 1 . (1) Let the linear equation to be solved be [ ] u B x A dt x d ] [ + = (2) where [A] is a NxN real constant matrix and [B] is a NxM real constant matrix and u is a M-tuple vector. The difficulty in solving (2) is that [A] is generally not a diagonal matrix and therefore the states x 1 , x 2 ,..x N are coupled together. Let us make a transformation from the x-space to a q-space using a transformation NxN transformation matrix [S], which is yet to be determined: [ ] q S x = (3) There is an inverse transformation by the inverse matrix [S]-1 : [ ] x S q 1 = (4) Substituting (3) in (2) u B q S A dt q d S ] [ ] ][ [ ] [ + = (5) Pre-multiplying (5) by [S]-1 u B S q S A S dt q d ] [ ] [ ] ][ [ ] [ 1 1 + = (6) Suppose we can find the matrix [S] such that ] [ ] ][ [ ] [ 1 = S A S (7) where = N ....
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- Spring '09