Eigen Values and Eigen Vectors for a real Matrix and example problems
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Step-I: To find the characteristic equation IA 11 — O Step-2: To solve the characteristic equation we get characteristic roots. They are called Eigen values. Step-3:To find Eigen vectors (A- 11 ) for the different values of 1 . PROBLEMS: 1. If -1, 1, 4 are the Eigen values of a matrix A of order 3 and , , [1,1, Solution: Eigen values are -1, 1, 4 The eigen vectors are 1] T are corresponding Eigen vectors, determine the matrix A. 0 1 1 2 1 1 1 1 1 The given set of eigen vectors are orthogonal in pair. Therefore the given matrix is a symmetric matrix. Therefore we use orthogonal transformation to Diagonalise that symmetric matrix. 2.Prove that the Eigen vectors corresponding to distinct Eigen values of a real Symmetric matrix are orthogonal. (Nov. 2002) Solution: For a real symmetric matrix A, the Eigen values are real. Let Xl and X be the Eigen vectors corresponding to distinct Eigen values & (real) (1)
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Similarly AX 2 12 X 2 (2) From (l) & (2) , Xl and X 2 are orthogonal 3. Find the Eigen values and Eigen vectors of the matrix Solution: 0 2 2 0 2 2 2 Let A 2 2 D3 1; D2 -12 0 45 To find the characteristic equation For the given matrix DI —2 + 1 + 0 211 45 0 The Eigen values are 5, 3, 3 6=-21; The Eigen vectors are given by (A Il)X 21 2 which is 6 2 For 1 7 5, 2 3 6
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Solving by cross multiplication method 24 For 2 3, 4 2 24 3 6 3 Solving by cross multiplication method — Which does not exist? + 2X2 3X3 0 If 0 The given matrix is not symmetric; therefore the Eigen vectors need not be orthogonal. Therefore we choose the Eigen vector which may dependent or independent of other Eigen vectors. Let 3 4. Find the Eigen values and eigen vectors of the matrix Solution: 2 20 10 30 10 13
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Let 2 20 10 30 10 13 For the given matrix DI — 1; D2 —2; D The characteristic equation is given by The Eigen values are given by 0, —1, 2 The Eigen vectors are given by (A Il)X = O 0 21 0 For 1=0, 20 10 10 Solving by cross multiplication method 20 4 For 0 1, 20 11 1 10 Solving by cross multiplication method 30 o For 1=2, 15 20 0 10 Solving by cross multiplication method
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o 12 24 5.Find the Eigen values and Eigen vectors of the matrix Solution: 2 2 3 I 32 2 1 3 0 . (May 2005) Let A 2 2 3 I 2 1 which is a symmetric matrix 3 For the given matrix DI = 12; D2 — 36; D The characteristic equation is given by The Eigen values are given by 8, 2, 2 The Eigen vectors are given by (A Il)X Forl 8, Solving by cross multiplication method 32 1212 +361 For 12 2, Solving by cross multiplication method
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o o . 4Xl 2X2 -k 2X3 — which is an impossible eigen vector. 0 If Xl = X2 2, To choose the eigen vector, remember we are dealing with a symmetric matrix, in which the eigen vectors are orthogonal in pairs. Let b be the third eigen vector which is orthogonal to Xl & X Similarly X2 L X 3 > a 2c—0 Solving the above two equations by cross multiplication method, we get 5
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