matrixeigenfacts - Selected Properties of Matrices...

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Page 1 of 1 Last Updated: May 9, 2008 2.0 Eigenvalues and Eigenvectors a. The eigenvalue problem is of the form Ax = λ x where the scalar λ is the eigenvalue and x is the (nonzero) eigenvector. b. If x is an eigenvector, then so is c x where c is any nonzero scalar. c. A symmetric matrix A will have n eigenvalues and n corresponding eigenvectors. Some eigenvalues may be repeated. d. Symmetric matrices always have real eigenvalues and real eigenvectors. Nonsymmetric matrices may have complex eigenvalues and eigenvectors. The number of eigenvectors in the nonsymmetric case may be less than n. e. The eigenvalues of A are the roots of the characteristic polynomial p(λ) = | A - λ I |=0. If the characteristic polynomial is defined as in Matlab: P(λ)=|λ I - A |=0, then p(λ) = ( -1) n * P(λ). f. (Gerschgorin Circle Theorem). All the eigenvalues of a matrix lie in the set union of circles in the complex plane. The centers are at the diagonals of the matrix and the radii of each circle is the sum of the absolute values of the off-diagonals in
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