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# final_review (dragged) 3 - 4 MA 36600 FINAL REVIEW Given an...

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4 MA 36600 FINAL REVIEW Given an n × n matrix A , consider an equation of the form A x = λ x where x is an n -dimensional vector and λ is a scalar. If this equation holds for some nonzero vector x and arbitrary scalar λ , we call x an eigenvector of A , and λ an eigenvalue of A . Since an eigenvalue x is a nontrivial solution to ( A λ I ) x = 0 , the matrix ( A λ I ) is singular. Hence λ is an eigenvalue for A if and only if det ( A λ I ) = 0. § 7.4: Basic Theory of Systems of First Order Linear Equations. Consider the general system of first order linear equations d dt x = P ( t ) x + g ( t ) . Say that we can find ( n + 1) matrix functions x (1) ( t ) , x (2) ( t ) , . . . , x ( n ) ( t ) and x ( p ) ( t ) such that i. The x ( k ) are solutions to the homogeneous equation d dt x ( k ) = P ( t ) x ( k ) , k = 1 , 2 , . . . , n. ii. The n × n matrix Ψ ( t ) = x 11 ( t ) x 12 ( t ) · · · x 1 n ( t ) x 21 ( t ) x 22 ( t ) · · · x 2 n ( t ) . . . . . . . . . . . . x n 1 ( t ) x n 2 ( t ) · · · x nn ( t ) where x ( k ) ( t ) = x 1 k ( t ) x 2 k ( t ) .
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