More Eigenvalues and Eigenvectors

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Lecture 16

• More Eigenvalues and Eigenvectors

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Lecture 16 Objectives

• For a given square matrix find
• The characteristic polynomial
• The eigenvalues, and their corresponding
  eigenvectors
• The algebraic multiplicity of each eigenvalue
• Use the eigenvalues and eigenvectors of a matrix
  to find repeated applications of the matrix
  transformation on a given vector
• Use Cayley Hamilton Theorem to find powers and
  inverses of matrices

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Example
Algebraically find all eigenvalues and
corresponding eigenvectors for the matrix
Note A has an eigenvalue ? iff Ax ?x for
some nonzero vector x iff (A ? ?I)x 0 for
some nonzero vector x iff det(A ? ?I) 0.
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Finding Eigenvalues and Eigenvectors

• To find the eigenvalues and corresponding
  eigenvectors for an n?n matrix A
• Solve the equation det(A ? ?I) 0 for ? and get
  (possibly) n values ?1, ?2,, ?n.
• For each ?i, solve the equation (A ? ?iI)x 0 to
  get all eigenvectors x corresponding to ?i.
• Example find all eigenvalues and corresponding
  eigenvectors for the matrix

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Notes and Definitions

• The function p(?) det(A ? ?I) is an n-degree
  polynomial called the characteristic polynomial
  of A, and the equation det(A ? ?I) 0 is called
  the characteristic equation of A.
• The characteristic polynomial can be factorized
  to p(?) (?1 ? ?)(?2 ? ?)(?n ? ?)
  giving (possibly) n values of ?.
• The algebraic multiplicity of ?i is its
  multiplicity in p(?).
• Example If p(?) (2 ? ?)3(4 ?)2(? ? 5), then
  the distinct eigenvalues are 2, ?4, and 5 with
  algebraic multiplicity 3, 2, and 1, respectively.

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Example

• For the matrix
• Find
• The characteristic polynomial of A.
• The eigenvalues of A, and their algebraic
  multiplicity
• The corresponding eigenvectors.
• (See also example 4.18 of Pooles text.)

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Properties of Eigenvalues and Eigenvectors

• Let A be a square matrix with eigenvalue ? and a
  corresponding eigenvector x, i.e. Ax ?x.
  Then
• An has an eigenvalue ?n with corresponding
  eigenvector x, i.e. Anx ?nx.
• If A is invertible, then ? ? 0, and A?1 has an
  eigenvalue ??1 with corresponding eigenvector x,
  i.e. A?1x ??1x.

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Repeated Application of Matrix Transformation
using Eigenvectors

• Suppose v1, v2,, vn are eigenvectors of A
  corresponding to the eigenvalues ?1, ?2,, ?n,
• and let x be a vector that is represented as a
  linear combination of these eigenvectors, i.e.
  x c1v1 c2v2 cnvn.
• Then we can easily calculate
• Akx c1Akv1 c2Akv2 cnAkvn
• c1?1kv1 c2?2kv2 cn?nkvn.

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Example

• Let
• Find
• The eigenvalues and the corresponding
  eigenvectors of A.
• A100x
• (See also example 4.21 of Pooles text.)

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Cayley-Hamilton Theorem

• Theorem If p(?) det(A ? ?I) is the
  characteristic polynomial of an n?n matrix A,
  then p(A) 0.
• Proof (if A has n independent eigenvectors) Let
  v be an eigenvector of A corresponding to an
  eigenvalue ?. Since Akv ?kv, we also get p(A)v
  p(?)v 0. Since A has n independent
  eigenvectors, this implies that p(A) 0.
• Example Verify the Cayley-Hamilton Theorem for
  the matrix

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Applications of the Cayley-Hamilton Theorem

• The Cayley-Hamilton Theorem can be used to find
• The Power and
• The Inverse
• of an n?n matrix A, by expressing these as
  polynomials in A of degree lt n.
• Example Use the Cayley-Hamilton Theorem to find
  A4 and A?1 where

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• Wafik