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expm_2009_10_26_01

# expm_2009_10_26_01 - 11 1 Matrix Exponential S Lall...

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Unformatted text preview: 11 - 1 Matrix Exponential S. Lall, Stanford 2009.10.26.01 11. Matrix Exponential Diagonalization Modal form Matrix exponential Sampling a continuous-time system Stability 11 - 2 Matrix Exponential S. Lall, Stanford 2009.10.26.01 Diagonalization suppose v 1 ,...,v n is a linearly independent set of eigenvectors of A R n n : Av i = i v i , i = 1 ,...,n express as A bracketleftbig v 1 v n bracketrightbig = bracketleftbig v 1 v n bracketrightbig 1 . . . n define T = bracketleftbig v 1 v n bracketrightbig and = diag( 1 ,..., n ) , so AT = T hence T 1 AT = T invertible since v 1 ,...,v n linearly independent similarity transformation by T diagonalizes A 11 - 3 Matrix Exponential S. Lall, Stanford 2009.10.26.01 Modal form Suppose A is diagonalizable by T . Define new coordinates by x = T x , so T x = AT x x = T 1 AT x in new coordinate system, system is diagonal x = x Called modal form ; trajectories consist of n independent modes, i.e. , x i ( t ) = e i t x i (0) if initial state x (0) is an eigenvector v , resulting motion is simple always on the line spanned by v for R , &amp;lt; , mode contracts or shrinks as t for R , &amp;gt; , mode expands or grows as t 11 - 4 Matrix Exponential S. Lall, Stanford 2009.10.26.01 Matrix exponential For a square matrix M , the matrix exponential is defined to be e M = I + M + M 2 2!...
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expm_2009_10_26_01 - 11 1 Matrix Exponential S Lall...

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