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notes7_3 - MA366 Sathaye Fast methods for some chapter 7...

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MA366 Sathaye Fast methods for some chapter 7 problems. We show some faster methods to quickly get a fundamental matrix for linear systems of differential equations. Two variable systems. 1. Basic formulas. We consider a system X 0 = AX where A is a 2 × 2 matrix with constant coefficients. Let A = a b c d . We note that its characteristic polynomial has the formula P A ( λ ) = λ 2 - λ ( a + d ) + ad - bc . We typically solve for eigenvalues by finding the roots of P A ( λ ) and depending on the nature of roots, we have different techniques for solving the system. 2. Fundamental Matrix For this discussion, we may even consider an n × n matrix A giving a system of n equations in n variables. A Fundamental matrix for the system X 0 = AX is defined to be an n × n matrix Ψ( t ) whose columns are n independent solutions of the system. This can be shown equivalent to the condition Ψ 0 ( t ) = A Ψ( t ) together with the condition that Ψ( t ) is invertible on an interval where the solutions are discussed. The Special Fundamental Matrix is given by Φ( t ) = exp ( At ) = I + At + A 2 t 2 / 2! + · · · Its columns are independent solutions of the system and have the added property that Φ(0) = I . Indeed, this extra condition identifies a Fundamental Matrix as a Special Fundamental Matrix. A general solution to the system X 0 = AX is then given by X ( t ) = Φ( t ) C where C is a column of arbitrary constants. In particular, X = Φ( t ) X (0) where X (0) = X (0) is the given set of initial conditions. We give direct methods to find this matrix Φ( t ). 3. Exponential shift. Consider a substitution X = exp( rt ) Y . Substitution in the equation X 0 = AX gives X 0 = exp( rt ) rY + exp( rt ) Y 0 = A exp( rt ) Y. By canceling exp( rt ) and rearranging, we see that Y 0 = ( A - rI ) Y. It is easy to see that the characteristic polynomial of A - rI is simply P A ( λ + r ). We shall use this substitution to simplify our characteristic polynomial. Finally, whatever special fundamental matrix is found for the system Y 0 = ( a - rI ) Y , we may multiply it by exp( rt ) to get the matrix for the system X 0 = AX . 1
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4. The Cayley-Hamilton Theorem. Suppose A is an n × n matrix with characteristic polynomial P A ( λ ) = ( - 1) n λ n + a 1 λ ( n - 1) + · · · + a ( n - 1) λ + a n then P A ( A ) = ( - 1) n A n + a 1 A ( n - 1) + · · · + a ( n - 1) A + a n I = 0 .
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