lesson27 - MA 511 Session 27 Eigenvalues and Eigenvectors Let A be a n n matrix Denition We say the number(real or complex is an eigenvalue of A if the

lesson27 - MA 511 Session 27 Eigenvalues and Eigenvectors...

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Unformatted text preview: MA 511, Session 27 Eigenvalues and Eigenvectors Let A be a n × n matrix. Definition: We say the number λ (real or complex) is an eigenvalue of A if the matrix A- λ I is sin- gular, that is, det( A- λ I ) = 0. In that case, the system ( A- λ I ) x = 0 has a non-trivial (nonzero) solution, e.g., there is a nonzero vector v such that Av = λv . Such a vector is called an eigenvector of A corresponding to the eigenvalue λ . First-Order Linear Homogeneous Systems of Ordinary Differential Equations: du 1 dt = a 11 u 1 + ··· + a 1 n u n · · · du n dt = a n 1 u 1 + ··· + a nn u n We are interested in nontrivial (nonzero) solutions. We let u ( t ) = ( u 1 ( t ) , . . . , u n ( t ) ) T and write the sys- tem in matrix form as du dt = Au. By analogy with the single equation, we seek expo- nential solutions u = e λt v , where v ∈ R n is a con- stant vector. Substituting in the equation, we obtain λe λt v = Ae λt v. Multiplying both sides by e- λt , we see that Av = λv , that is ( A- λ I ) v = 0 . Thus a nonzero solution u = e λt v exists exactly when λ is an eigenvalue of A , and v is an eigenvector. Characteristic polynomial. p ( λ ) = det( A- λ I ) is a polynomial of degree n , the characteristic polynomial of A . The eigenvalues of A are the roots of p . The total number of eigen- values, counted with multiplicities, is n . When A is real, p ( λ ) may have complex roots. If λ = a + bi is an eigenvalue, then so is its complex conjugate ¯ λ = a- bi ....
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