5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors.

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Presentation transcript:

5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors

For eigenvalue eigenvector

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 ?

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

Definition The set of all solutions of is called the eigenspace of A corresponding to. Note: 1.Eigenspace of A is a null space of. 2. Eigenspace of A is a subspace of for matrix A.

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

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

Example: Find the eigenvalues of

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