Engineering Applications

1
Engineering Applications

• Dr. Darrin Leleux
• Lecture 10 Eigenvalues and Eigenvectors
• Chapter 4

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Introduction

• Number or Independent Eigenvectors
• 4.3 Matrix Eigenvalue Theorems
• 4.4 Complex Vectors and Matrices
• 4.5 MATLAB Commands for Eigenvectors
• 4.6 Matrix Calculus
• 4.7 Similar and Diagonalizable Matrices
• 4.8 Special Matrices and their Eigenvalues
• 4.9 Applications to Differential Equations
• Homework 4

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Number of Independent Eigenvectors (pg. 167)

• Theorem 4.1 Distinct Eigenvalues
• Eigenvectors corresponding to the distinct
  eigenvalues of a matrix are linearly independent
• Theorem 4.2 Repeated Eigenvalues
• If ? is an eigenvalue of multiplicity k of an n x
  n matrix A, then the number of independent
  eigenvectors N of A associated with ? is given by
• N n rank(A - ?I)
• and, furthermore, 1 lt N lt k.

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Example 4.6, pg. 168Matrix Eigenvalue Example
Consider
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4.3 Matrix Eigenvalue Theorems

• ? is an eigenvalue of A if and only if A - ?I
  0
• If ? is an eigenvalue of A, any nontrivial
  solution to (A - ?I)x 0 is an eigenvector of A
  corresponding to eigenvalue ?.
• The determinant of A is the product of its
  eigenvalues so that det(A) ?1 ?2? ? ? ?n
• A is singular if and only if it has an eigenvalue
  of zero
• The sum of the diagonal elements of A (called the
  trace) of A is equal to the sum of its
  eigenvalues
• The eigenvalues of a triangular matrix are its
  diagonal entries

Example 4.7, pg. 171
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4.4 Complex Vectors and Matrices

• Eigenvalues and Eigenvectors can be complex even
  though the matrix has real elements
• From Ch. 2, ltz1,z2gt (z1)Tz2 which is called
  the conjugate transpose
• The superscript H designates (z)T zH called
  the Hermitian transpose
• The same definition applies with matrices

Example 4.8, pg. 173
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4.5 MATLAB Commands for Eigenvectors

• trace returns sum of diagonal elements
• eig(A) eigenvalues of A
• V,Deig(A) Solves for eigenvectors and
  eigenvalues of A
• V is a matrix whose columns are eigenvectors with
  norm equal to 1
• D is a diagonal matrix whose elements are the
  eigenvalues (AV VD)
• poly Characteristic polynomial for A - ?I
• roots Roots of polynomial p(x)
• sym,syms symbolic results
• eig(sym(As))
• poly(As)

Example 4.9, pp. 174-175
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Numerical Computation

• QR decomposition is a method of computing the
  eigenvalues and eigenvectors
• Modification of the Gram-Schmidt process for
  creating an orthonormal set of vectors in Ch.2
• Theorem 4.3 Uniqueness of QR decomposition
• Let A be an m x n matrix with m gt n and linearly
  independent columns. Then there exists a unique
  m x n matrix Q such that QTQ D
  diag(d1,,dn), dk gt 0,for k 1, , n, and a
  unique upper-triangular matrix R, with rkk
  1, k 1, ,n,such that A QR

Example 4.10, pp. 177-178
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Numerical Errors

• Matrices without a full set of linearly
  independent eigenvectors are referred to as
  defective matrices.
• Defective matrices are not diagonalizable
• If eigenvalues are repeated, the matrix A may or
  may not have n independent eigenvectors.

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4.6 Matrix Calculus

• Integration and Differentiation
• If each of the elements of aij(t) is a
  differentiable function of time, the A(t) is the
  derivative of each of the elements with respect
  to time.
• If each of the elements of aij(t) is a integrable
  function of time, the A(t) is the integral of
  each of the elements with respect to time.
• Matrix polynomials
• Pn(A) anAn an-1An-1a0I

Manually you can do this for low powers of n, but
the Better way is with the Cayley-Hamilton theorem
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Exponentials
Again, manually you can do this for low powers of
k, but the better way is with the Cayley-Hamilton
theorem
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MATLAB Matrix Functions

• expm Matrix exponential of A
• funm Functions of a matrix A
• logm Inverse function of expm
• sqrtm Square root of a matrix, XXA
• int element-by-element integration
• of a symbolic matrix

Example 4.11, pg. 181
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Cayley-Hamilton Theorem

• Every square matrix A satisfies its own
  characteristic equation A-?I 0, so that if
• ?n an-1?n-1 a1? a0 0
• Then
• An an-1An-1 a1A a0I 0
• Furthermore, when the positive form of the
  eigenvalues is used from eq 4.18 on pg. 162, the
  product of the eigenvalues is (-1)na0 and it
  follows that det(A) (-1)na0

Example 4.12 on pg. 184 Example 4.13 on pg.
185-186
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4.7 Similar and Diagonalizable

• Two n x n matrices A and B are said to be similar
  if there exists an invertible n x n matrix P such
  that A P-1BP
• This is called the similarity transformation
• A matrix is diagonalizable if it is similar to a
  diagonal matrix
• Theorem 4.5 Independent eigenvectors
• A n x n matrix A is diagonalizable if and only if
  it possesses n linearly independent eigenvectors

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Diagonalizable Matrices

• When A has n linearly independent eigenvectors
  x1, x2, , xn, we can form the modal matrix
• M x1, x2, , xn
• Additionally, we can form the diagonal matrix
  which has its eigenvalues as the diagonal
  elements. This is called the spectral matrix of
  A
• Finally, AM MD from pg. 188 thus A MDM-1
• A is similar to D, and M M-1 are are used to
  diagonalize A
• And Am MDmM-1

Example 4.14, pg. 189
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4.8 Special Matrices and Their Eigenvalues

• A real symmetric matrix is one where A AT. This
  also includes real diagonal matrices.
• The eigenvalues and eigenvectors of such matrices
  are always real
• Eigenvectors corresponding to distinct
  eigenvalues are orthogonal
• Any symmetric matrix is diagonalizable

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Real Symmetric Matrices

• Theorem 4.6 - Orthogonal diagonalization
• For every n x n real symmetric matrix A, there
  exists an n x n real orthogonal matrix Q such
  that Q-1AQ QTAQ ?where ? is a diagonal
  matrix
• When this principle holds, the matrix A is said
  to be orthogonally diagonalizable. This is also
  called the principal axis theorem in geometry or
  mechanics

Example 4.14, pg. 191
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Covariance Matrices

• Matrices can represent position uncertainty
  ellipses (2D) or ellipsoids (3D)

3D
2D
BOTH are symmetrical
HOWEVER off-diagonal terms mean the ellipse or
ellipsoid is rotated out of standard position,
i.e. it is NOT aligned with the UVW axes
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Covariance Matrices
Ellipse can be represented in quadratic form

• How do matrices represent ellipses or ellipsoids

The equation would be
(constant)
Off-diagonal xy components of the equation
indicate the ellipse is rotated
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Covariance Matrices

• To rotate the ellipse back into standard position

Find the eigenvalues of the ellipse
The equation would be
Use Principal Axis Theorem for R2 Eigenvectors
form the basis for the new rotated coordinate
system
Equation in standard notation for C 96
Length of semi-major and semi-minor axes is
Eigenvector Equation
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Hermitian and Unitary Matrices

• The matrix with complex elements that plays the
  role of a symmetrical matrix is called Hermitian
• These matrices are equal to their conjugate
  transpose, i.e. AH A
• A Hermitian matrix has real eigenvalues
• Eigenvectors of different eigenvalues are
  orthogonal to one another
• It is also diagonalizable
• The complex equivalent of an orthongonal matrix
  is the unitary matrix where UH U-1.
• Thus, ? U-1AU UHAU

Example 4.16, pg. 192
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Summary

• Real matrices
• Symmetric ATA Real eigenvalues
• Skew AT -A Imaginary or zero eigenvalues
• Orthogonal QTQ-1 All ?i 1
• Complex matrices
• Hermitian AHA Real eigenvalues
• Skew Hermitian AH-A Imaginary or zero
  eigenvalues
• Unitary UHU-1 All ?i 1

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4.9 Applications to Differential Equations

• The idea of linear algebraic equations in matrix
  form can be extend to differential equations
• Consider the system of differential
  equationsHas the solution

Example 4.17, pg. 195
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System of Differential Equation, Ex. 4.18, Figure
4.4