Linear Algebra Chapter 6

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Linear Algebra Chapter 6

• Linear Algebra with Applications
• -Gareth Williams
• Br. Joel Baumeyer, F.S.C.

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Definition Eigenvalue Eigenvector

• The Characteristic Polynomial of Anxn
• A - ?I
• The Characteristic Equation of Anxn
• A - ?I 0

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Theorem 6.1
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Similar Matrices

• Definition Let A and B be square matrices of
  the same size. B is said to be similar to A if
  there exists an invertible matrix C such that B
  C-1AC. The transformation of the matrix A into
  the matrix B in this manner is called a
  similarity transformation.

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Theorem 6.3

• Theorem 6.2 is in the optional section 6.2 and
  not covered.

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Diagonalizable Matrix

• A square matrix A is said to be diagonalizable if
  there exists a matrix C such that D C-1AC is a
  diagonal matrix.

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Theorem 6.4

• Let A be an n?n matrix.
• (a) If A has n linearly independent eigenvectors,
  it is diagonalizable. The matrix C whose columns
  consist of n linearly independent eigenvectors
  can be used in a similarity transformation C-1AC
  to give a diagonal matrix d. The diagonal
  elements of D will be the eigenvalues of A.
• (b) If A is diagonalizable, then it has n
  linearly independent eigenvectors.

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Powers of Diagonalizable Matrices

• If A is similar to a diagonal mamatrix D under
  the transformation C-1AC, then it can be shown
  that Ak CDkC-1.
• Demonstration
• Dk (C-1AC)k (C-1AC) (C-1AC) (C-1AkC)
• and reversing gives Ak CDkC-1

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Theorem 6.5

• Let a be an nxn symmetrix matrix.
• a) All the eigenvalues of A are real numbers.
• b) the dimension of an eigenspace of A is the
  multiplicity of the eigenvalue as a root of the
  characteristic equation.
• c) The eigenspaces of A are orthogonal.
• d)A has n linearly independent eigenvectors.

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Orthogonally Diagonalizable

• Definition
• A square matrix A is said to be orthogonally
  diagonalizable if there exists an ortholgonal
  matrix C such that D CtAC is a diagonal matrix.

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Theorem 6.6

• Let a be a square matrix. A is orthog-onally
  diagonalizable if and only if it is a symmetric
  matrix.