Chapter 7 Eigenvalues and Eigenvectors

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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

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• 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.
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• Eigenvalues and eigenvectors of linear
  transformations

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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.

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7.3 Symmetric Matrices and Orthogonal
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• Note Theorem 7.7 is called the Real Spectral
  Theorem, and the set of eigenvalues of A is
  called the spectrum of A.

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

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7.4 Applications of Eigenvalues and
Eigenvectors
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• If A is not diagonal
• -- Find P that diagonalizes A

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• Quadratic Forms

and are eigenvalues of the matrix
matrix of the quadratic form
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