Page 1 Review of Eigenvectors and Eigenvalues from CliffsNotes Online http://www.cliffsnotes.com/study_guide/Deter mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId- 20804.html 1 Page 2 Review of Eigenvectors and Eigenvalues from CliffsNotes Online http://www.cliffsnotes.com/study_guide/Deter mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId- 20804.html 1 Definition The eigenvectors x and eigenvalues ? of a matrix A satisfy Ax = ?x If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant. The equation can be rewritten as (A - ?I) x = 0, where I is the n x n identity matrix. 2 Page 3 Review of Eigenvectors and Eigenvalues from CliffsNotes Online http://www.cliffsnotes.com/study_guide/Deter mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId- 20804.html 1 Definition The eigenvectors x and eigenvalues ? of a matrix A satisfy Ax = ?x If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant. The equation can be rewritten as (A - ?I) x = 0, where I is the n x n identity matrix. 2 Computing Eigenvalues Since x is required to be nonzero, the eigenvalues must satisfy det(A - ?I) = 0 which is called the characteristic equation. Solving it for values of ? gives the eigenvalues of matrix A. 3 Page 4 Review of Eigenvectors and Eigenvalues from CliffsNotes Online http://www.cliffsnotes.com/study_guide/Deter mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId- 20804.html 1 Definition The eigenvectors x and eigenvalues ? of a matrix A satisfy Ax = ?x If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant. The equation can be rewritten as (A - ?I) x = 0, where I is the n x n identity matrix. 2 Computing Eigenvalues Since x is required to be nonzero, the eigenvalues must satisfy det(A - ?I) = 0 which is called the characteristic equation. Solving it for values of ? gives the eigenvalues of matrix A. 3 2 X 2 Example A = so A - ?I = 1 -2 1 - ? -2 3 -4 3 -4 - ? det(A - ?I) = (1 - ?)(-4 - ?) – (3)(-2) = ? 2 + 3 ? + 2 Set ? 2 + 3 ? + 2 to 0 Then = ? = (-3 +/- sqrt(9-8))/2 So the two values of ? are -1 and -2. 4 Page 5 Review of Eigenvectors and Eigenvalues from CliffsNotes Online http://www.cliffsnotes.com/study_guide/Deter mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId- 20804.html 1 Definition The eigenvectors x and eigenvalues ? of a matrix A satisfy Ax = ?x If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant. The equation can be rewritten as (A - ?I) x = 0, where I is the n x n identity matrix. 2 Computing Eigenvalues Since x is required to be nonzero, the eigenvalues must satisfy det(A - ?I) = 0 which is called the characteristic equation. Solving it for values of ? gives the eigenvalues of matrix A. 3 2 X 2 Example A = so A - ?I = 1 -2 1 - ? -2 3 -4 3 -4 - ? det(A - ?I) = (1 - ?)(-4 - ?) – (3)(-2) = ? 2 + 3 ? + 2 Set ? 2 + 3 ? + 2 to 0 Then = ? = (-3 +/- sqrt(9-8))/2 So the two values of ? are -1 and -2. 4 Finding the Eigenvectors Once you have the eigenvalues, you can plug them into the equation Ax = ?x to find the corresponding sets of eigenvectors x. 1 -2 x 1 = -1 x 1 so 3 -4 x 2 x 2 x 1 – 2x 2 = -x 1 3x 1 – 4x 2 = -x 2 (1) 2x 1 – 2x 2 = 0 (2) 3x 1 – 3x 2 = 0 These equations are not independent. If you multiply (2) by 2/3, you get (1). The simplest form of (1) and (2) is x 1 - x 2 = 0, or just x 1 = x 2 . 5Read More

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