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Chapter 7 Eigenvalues and Eigenvectors

Chapter 7 Eigenvalues and Eigenvectors. 7.1 Eigenvalues and eigenvectors. Eigenvalue problem: If A is an n  n matrix, do there exist nonzero vectors x in Rn such that A x is a scalar multiple of x. Note:. (homogeneous system). Characteristic polynomial of A  M n  n :.

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Chapter 7 Eigenvalues and Eigenvectors

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  1. Chapter 7 Eigenvalues and Eigenvectors 7.1 Eigenvalues and eigenvectors • Eigenvalue problem: If A is an nn matrix, do there exist nonzero vectors x in Rn such that Ax is a scalar multiple of x

  2. Note: (homogeneous system) • Characteristic polynomial of AMnn: If has nonzero solutions iff . • Characteristic equation of A:

  3. Notes: (1) If an eigenvalue 1 occurs as a multiple root (k times) for the characteristic polynomial, then1 has multiplicity k. (2) The multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace.

  4. Eigenvalues and eigenvectors of linear transformations:

  5. 7.2 Diagonalization • Diagonalization problem: For a square matrix A, does there exist an invertible matrix P such that P-1AP is diagonal? • Notes: • (1) If there exists an invertible matrix P such that , • then two square matrices A and B are called similar. • (2) The eigenvalue problem is related closely to the • diagonalization problem.

  6. 7.3 Symmetric Matrices and Orthogonal

  7. Note: Theorem 7.7 is called the Real Spectral Theorem, and the set of eigenvalues of A is called the spectrum of A.

  8. Note: A matrix A is orthogonally diagonalizable if there exists an orthogonal matrix P such that P-1AP = D is diagonal.

  9. 7.4 Applications of Eigenvalues and Eigenvectors

  10. If A is not diagonal: • -- Find P that diagonalizes A:

  11. Quadratic Forms and are eigenvalues of the matrix: matrix of the quadratic form

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