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5. Eigenvalues and Eigenvectors

5. Eigenvalues and Eigenvectors. 5.1 Eigenvectors and Eigenvalues. For. eigenvalue. eigenvector. Definition

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5. Eigenvalues and Eigenvectors

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  1. 5. Eigenvalues and Eigenvectors 5.1 Eigenvectors and Eigenvalues

  2. For eigenvalue eigenvector

  3. Definition An eigenvector of an matrix A is a nonzero vector x such that for some scalar . A scalar is called an eigenvalue of A if there is a nontrivial solution x of ; such an x is called an eigenvector corresponding to . Example: Are u and v eigenvectors of A?

  4. Example: Show that 7 is an eigenvalue of and find the corresponding eigenvectors.

  5. Definition The set of all solutions of is called the eigenspace of A corresponding to . • Note: • Eigenspace of A is a null space of . • 2. Eigenspace of A is a subspace of for matrix A.

  6. Example: Find the basis for the corresponding eigenspace of to eigenvalue 2.

  7. Theorem 1 The eigenvalues of a triangular matrix are the entries on its main diagonal.

  8. Example: Find the eigenvalues of

  9. Theorem 2 If are eigenvectors that correspond to distinct eigenvalues of an matrix A, then the set is linearly independent.

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