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Unformatted text preview: ECSE 462 Lecture 19 Revisions of Linear Algebra Consider an Ntuple vector âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¦ âŽ¤ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ£ âŽ¡ = âˆ’ 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 Mtuple 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 xspace to a qspace 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 âˆ’ = (4) Substituting (3) in (2) u B q S A dt q d S ] [ ] ][ [ ] [ + = (5) Premultiplying (5) by [S]1 u B S q S A S dt q d ] [ ] [ ] ][ [ ] [ 1 1 âˆ’ âˆ’ + = (6) Suppose we can find the matrix [S] such that ] [ ] ][ [ ] [ 1 Î› = âˆ’ S A S (7) where âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¦ âŽ¤ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ£ âŽ¡ = Î› N Î» Î» Î» ....
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This note was uploaded on 02/23/2010 for the course ECSE 462 taught by Professor Bontekooi during the Spring '09 term at McGill.
 Spring '09
 BonTekOoi

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