# Eigen - ME 210 Applied Mathematics for Mechanical Engineers...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Eigenvalues and Eigenvectors In a large number of physical and technical problems, the system of linear equations of the type a 11 x 1 + a 12 x 2 + … + a 1n x n = λ x 1 a 21 x 1 + a 22 x 2 + … + a 2n x n = λ x 2 …………………………………… ……………………………………. a n1 x 1 + a n2 x 2 + … + a nn x n = λ x n is encountered where λ is a parameter. In matrix form: [A] [x] = λ [x]

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Obviously, is a solution of but it is trivial. The values of λ for which the system has solutions (other than the trivial) are called eigenvalues (characteristic values) of the matrix A. Each solution of the system corresponding to an eigenvalue is called an eigenvector of the matrix A. The set of all eigenvectors is called the spectrum of A.   x = 0 [A] [x] = λ [x] [A] [x] = λ [x] Take product of both sides with identity matrix: [I] [A] [x] = [I] λ [x] => [I] [A] [x] = λ [I] [x] => [A] [x] = λ [I] [x] Re-arrange: { A – λ [I] } [x] = [0]
ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Represents a homogeneous system of n linear equations in n unknowns. This has a non-trivial solution iff 11 12 1 21 22 2 12 D( ) det{[A] 0   n n n n nn a a a a a a a a a D(λ) is called as the characteristic determinant of matrix [A]. When developed, it gives a polynomial of degree n in λ, called as the characteristic polynomial of matrix [A]. The equation D(λ)=0 is called as the characteristic equation of matrix [A]. The roots λ 1 , λ 2 , …, λ n of the characteristic equation are called as Characteristic Values or Eigenvalues of matrix [A]. { A – λ [I] } [x] = [0]

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 There exists as many eigenvalues for each square matrix as its size. When any of these eigenvalues is used in set of homogeneous linear equations, it will be possible to obtain a corresponding non-trivial solution for [x]. The vectors so- obtained are called as the eigenvectors of matrix [A].
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## Eigen - ME 210 Applied Mathematics for Mechanical Engineers...

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