CLASS NOTES FOR LINEAR MATH

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CLASS NOTES FOR LINEAR MATH

• Section 4.2
• SHURS THEOREM AND THE EIGENVALUE PROBLEM FOR
  SYMMETRIC MATRICES

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THEOREM 1

• Let A be a symmetric matrix
• a. All eigenvalues of A are real
• b. Eigenvectors associated with (distinct)
  eigenvalues are orthogonal.

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THEOREM 2
Let A be nxn. A has real eigenvalues if and only
if there is a matrix P such that P-1AP PTAP
T The columns of P form an orthogonal set of
eigenvectors of A.
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THEOREM 3
If A is nxn with eigenvalues ?1, ..., ?n (listing
duplicates) and q(x) is a polynomial then B
q(A) has eigenvalues q(?1), ..., q(?n).
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STEP BY STEP PROCEDURE

• Suppose AT A. To find P so that PPT I and
  P-1AP is diagonal
• Solve det (?I-A) 0 and list with their
  multiplicities.
• for the solution of multiplicity 1, find an
  eigenvector and normalize.
• For the solutions of multiplicity gt 1, find k
  linearly independent eigenvectors and use
  Gram-Schmidt.
• Write your result as columns of P
• P-1AP will be diagonal.

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End of Section 4.2