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# MLT - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.B Matrix Laplace Transform Method Mon, 08/Dec c 2003, Art Belmonte Summary The Last Hurrah This is it campers: the Grand Finale, where we marry Laplace transform techniques with matrix methods for linear systems! It’s a little ditty we call the Matrix Laplace Transform Method . The Setting With t as an independent scalar variable, let A be an n × n constant matrix, f an n × 1 vector function of t , x ( t ) a function of t , and x 0 an n × 1 constant vector. Also, let s be a scalar variable. Some Definitions Denote the Laplace transform of x ( t ) by ˆ x ( s ) . It is a column vector whose elements are the respective Laplace transforms of the components x 1 ( t ), . . . , x n ( t ) of x ( t ) . In a similar manner, let L { f ( t ) } = ˆ f ( s ) . More briefly, we write L { x } = ˆ x and L { f } = ˆ f . Analogously, for a matrix M = M ( t ) , we define L { M } = ˆ M to be the matrix whose elements are the respective Laplace transforms of the components m ij ( t ) of M ( t ) . Inverse Laplace transforms of vectors or matrices are similarly defined. Moreover, the familiar

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MLT - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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