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Unformatted text preview: 4 MA 36600 FINAL REVIEW • Given an n × n matrix A, consider an equation of the form Ax = λx 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 x = P(t) x + g(t). dt 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 (k ) x = P(t) x(k) , k = 1, 2, . . . , n. dt ii. The n × n matrix x11 (t) x12 (t) · · · x1n (t) x1k (t) x21 (t) x22 (t) · · · x2n (t) x2k (t) Ψ(t) = where x(k) (t) = . . . . .. . . . . . . . . . xn1 (t) xn2 (t) · · · xnn (t) xnk (t) is nonsingular for all t. iii. x(p) is a particular solution to the nonhomogeneous equation d (p) x = P(t) x(p) + g(t). dt Then the general solution to the nonhomogeneous equation is in the form x(t) = c1 x(1) (t) + c2 x(2) (t) + · · · + cn x(n) (t) + x(p) (t) ￿ ￿ for constants ck . We call x(1) (t), x(2) (t), . . . , x(n) (t) a fundamental set of solutions for the homogeneous equation x￿ = P(t) x. ￿ ￿ • Let x(1) (t), x(2) (t), . . . , x(n) (t) be a a set of solutions to the homogeneous equation – not necessarily a fundamental set. Define their Wronskian as the determinant x11 (t) x12 (t) · · · x1n (t) x21 (t) x22 (t) · · · x2n (t) ￿ ￿ W x(1) , . . . , x(n) (t) = det . . . . .. . . . . . . . xn1 (t) xn2 (t) · · · xnn (t) Abel’s Formula asserts that there exists a constant C such that ￿ ￿t ￿ ￿ (1) ￿ ( n) W x , ..., x (t) = C · exp tr P(τ ) dτ in terms of the trace of an n × n matrix ￿ ￿ tr pij (t) = p11 (t) + p22 (t) + · · · + pnn (t). • Say i. ii. iii. iv. v. vi. ￿ (1) ￿ x (t), x(2) (t), . . . , x(n) (t) is a set of solutions. The following are equivalent: ￿ (1) ￿ x (t), x(2) (t), . . . , x(n) (t) is a fundamental set of solutions. ￿ (1) ￿ x (t), ￿ (2) (t), ￿ . . , x(n) (t) is a linearly independent set. x . Ψ(t0 ) =￿ xij (t￿ ) is a nonsingular matrix for some t0 . 0 Ψ(t) = xij (t) is a nonsingular matrix for all t. ￿ ￿ W x(1) , . . . , x(n) (t0 ) ￿= 0 for some t0 . ￿ (1) ￿ W x , . . . , x(n) (t) ￿= 0 for all t. ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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