Orthogonal Diagonalization

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Section 7.3

• Orthogonal Diagonalization

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TWO PROBLEMS

• Orthonormal Eigenvector Problem Given an nn
  matrix A, does there exist an orthonormal basis
  for Rn consisting of eigenvalues of A?
• Orthonormal Diagonalization Problem Given an
  nn matrix A, does there exist an orthogonal
  matrix P such that the matrix P-1AP  PTAP is
  diagonal? If there is such a matrix, then A is
  said to be orthogonally diagonalizable and P is
  said to orthogonally diagonalize A.

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CONDITIONS FOR ORTHOGONAL DIAGONALIZABILITY
Theorem 7.3.1 If A is an nn matrix, then the
following are equivalent. (a) A is orthogonally
diagonalizable. (b) A has an orthonormal set of n
eigenvectors. (c) A is symmetric.
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SYMMETRIC MATRICES
Theorem 7.3.2 If A is a symmetric matrix,
then (a) The eigenvalues of A are real
numbers. (b) Eigenvectors from different
eigenspaces are orthogonal.
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DIAGONALIZATION OF SYMMETRIC MATRICES
Step 1 Find a basis for each eigenspace of
A. Step 2 Apply the Gram-Schmidt process to each
of these bases to obtain an orthonormal basis for
each eigenspace. Step 3 Form the matrix P whose
columns are the basis vectors constructed in Step
2 this matrix orthogonally diagonalizes A.