Transcription of Fundamental Matrices, Matrix Exp & Repeated Eigenvalues ...
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Fundamental Matrices, Matrix Exp & Repeated Eigenvalues ...
Fundamental Matrices, Matrix Exp &. Repeated Eigenvalues Sections & Given Fundamental solutions G. G G dx G. x1 ,.., xn of the ODE = Ax dt we put them in an nxn Matrix G G. , (t ) = ( x1 ,.., xn ). with each of the solution vectors being a column. We call (t) a Fundamental Matrix for the system of ODEs. Example. G. dx 1 2 G. = x dt 2 1 . 1 2 . det = (1 ) 2. 4. 2 1 . = 2 2 3 = ( 3)( + 1). 1. 1 2 . So the Eigenvalues of the Matrix A= 2 1 .. in our ODE are =3,-1. The corresponding eigenvectors are found by solving (A- I)v=0 using Gaussian elimination. We find that 1 . the eigenvector for eigenvalue 3 is: v= . 1 . 1 . w= . the eigenvector for eigenvalue -1 is: 1 . So the corresponding solution vectors for our ODE. system are G 3t 1 G t 1 . u1 = e , u2 = e . 1 1 . Our Fundamental Matrix is: G G e 3t e . t (t ) = ( u1 u 2 ) = 3t t . e e . 2. The general solution is G G G c1 . y (t ) = c1u1 + c2u2 = (t ) . c2 . If you need to find c1,c2 satisfying some initial condition like y(t0)=b, then you need to solve G c1 b1.
Fundamental Matrices, Matrix Exp & Repeated Eigenvalues – Sections 7.7 & 7.8 Given fundamental solutions we put them in an nxn matrix , with each of the solution vectors being a column. We call Ψ(t) a fundamental matrix for the system of ODEs. Example. 2 2 12 21 12 det (1 ) 4 21 23( 3)( 1) dx x dt λ λ λ λλ λ λ ...
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