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Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors. Eigenvalues and Eigenvectors. The vector x is an eigenvector of matrix A and λ is an eigenvalue of A if: Ax= λ x Eigenvalues and eigenvectors are only defined for square matrices (i.e., m = n )

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Eigenvalues and Eigenvectors

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  1. Eigenvalues and Eigenvectors

  2. Eigenvalues and Eigenvectors • The vector xis an eigenvector of matrix Aand λ is an eigenvalue of A if: Ax= λx • Eigenvalues and eigenvectors are only defined for square matrices (i.e., m = n) • Eigenvectors are not unique (e.g., if λis an eigenvector, so is k λ) • Zero vector is a trivial solution to the eigenvalue equation for any number λ and is not considered as an eigenvector. • Interpretation: the linear transformation implied by A cannot change the direction of the eigenvectors λ, but change only their magnitude. (assume non-zero x)

  3. We summarize the computational approach for determining eigenpairs (, x) (eigenvalues and eigen vector) as a two-step procedure: Example: Find eigenpairs of Step I. Find the eigenvalues.

  4. The eigenvalues are Step II. To find corresponding eigenvectors we solve (A - iIn)x = 0 for i = 1,2 Note: rref means row reduced echelon form.

  5. Computing λ and v • To find the eigenvalues λ of a matrix A, find the roots of the characteristic polynomial : Ax= λx Example:

  6. Example: Find eigenvalues and eigen vectors of The characteristic polynomial is

  7. Example: Find the eigenvalues and corresponding eigenvectors of The characteristic polynomial is Its factors are So the eigenvalues are 1, 2, and 3. Corresponding eigenvectors are

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