Engineering Applications

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Engineering Applications

• Dr. Darrin Leleux
• Lecture 8 Matrices
• Chapter 3

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Introduction

• 3.5 Orthogonal and Triangular Matrices
• 3.6 Systems of Linear Equations
• 3.7 MATLAB Matrix Functions
• 3.8 Linear Transformations
• Homework 3

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Orthogonal and Triangular Matrices

• An orthogonal matrix Q is a square matrix that
  has the property
• Q-1 QT
• Thus the following is true QTQQQTI
• Good example of an orthogonal matrix is

Thus,
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Theorem 3.1

• The n x n matrix Q is orthogonal if and only if
  the columns (and rows) of Q form an orthonormal
  system

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Orthogonal Transformations

• Multiplication by an orthogonal matrix preserves
  the length of a vector and the value of the inner
  product

Thus it follows that,
pg 121
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Upper and Lower Triangular Matrices

• A matrix is upper triangular if all its elements
  below the diagonal are zero
• A matrix is lower triangular if all its elements
  above the diagonal are zero
• A matrix is diagonal if all it elements not on
  the diagonal are zero
• Pg 122 Theorems
• Theorem 3.2 det(A) a11a22ann
• Theorem 3.3 det(AB) det(A) det(B)

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LU Factorization

• Suppose the matrix A can be factored into the
  product of a lower triangular matrix L and an
  upper triangular matrix U so that A LU
• det(A)det(L) det(U)
• Thus (A)-1 (LU)-1 U-1L-1
• Example 3.11, pg. 123
• Doolittle method
• MATLAB command lu command performs LU
  decomposition

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Systems of Linear Equations

• 10I1 9I2 0I3 100
• -9I1 20I2 - 9I3 0
• 0I1 9I2 15I3 0

The system of equations can be written in the
form Ax b
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Terminology

• A solution is a set of n scalars x1, x2, , xn
  that satisfy the equations.
• A linear system of equations is consistent if it
  has a solution
• If there is no solution, the system is called
  inconsistent

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Possible states of a system of linear equations

• Consistent, with a unique (one) solution
• Consistent, with infinitely many possible
  solutions
• Inconsistent, with no solutions

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Terminology (continued)

• The linear system Ax 0 is called homogeneous,
  i.e. always has 0 has a solution
• If b ? 0, then it is called a nonhomogenous
  system
• If n gt m, the system has more unknowns than
  equations and is called underdetermined
• If n lt m, the system has more equations than
  unknowns and is called overdetermined
• To solve overdetermined systems, use methods such
  as least-squares treated in Chapter 7
• For underdetermined systems there are an infinite
  number of solutions
• Example 3.12, pg. 126

n of variables m of equations
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Solution by Matrix Inverse

• Let A be an n x n matrix. Then, the system Ax
  b has a unique solution if and only if A is
  invertible (nonsingular).
• x A-1 b
• Theorem 3.4
• Using the inverse involves more computations and
  potentially more roundoff error
• Example 7x21, using matrix inverse is akin to
  solving this by x 7-1 21 0.142857 21
  2.99997 instead of x 3 in a hypothetical
  machine
• Direction solutions of systems of equations are
  better than solving for the inverse

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Solution by Elementary Row Operations (Gaussian
Elimination)

• Form the augmented matrix with b as the fourth
  column
• Replace a row of the matrix by a nonzero
  multiple of the row
• Replace a row by the sum of that row and a
  multiple of another row
• Interchange two rows of the matrix
• Theorem 3.5 Equivalence of systems, pg 128
• Forms a reduced matrix

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Example 3.13
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Rank of a Square Matrix

• The rank of a matrix A is the number of linearly
  independent rows (or columns)
• Remember from Ch. 2, the determinant was used to
  determine linear independence
• The rows of a non-singular matrix are linearly
  independent
• rank can also be defined as the number of
  non-zero rows of the reduced matrix
• Also applies to non-square matrices

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Rank Theorems

• Theorem 3.6 Rank of nonsingular matrix
• The n x n matrix A is nonsingular if and only if
  the rank of A is n
• Theorem 3.7 Solution and rank of augmented
  matrix
• The nonhomogenous system of equations Axb has a
  solution if and only if rank(A) rank( Ab )
• Theorem 3.8 Inverse Matrix
• If an n x n matrix A can be converted to the n x
  n identity matrix I by a sequence of elementary
  operations, then the inverse A-1 is equal to the
  result of applying the same sequence of
  elementary operations to I

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MATLAB Matrix Functions

• det Determinant
• inv Inverse
• rank Rank
• rref (row echelon form) Reduced Matrix
• rrefmovie step-by-step reduction
• A\b Solve Ax b
• lu LU decomposition

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EXAMPLE 3.14
pg 132
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Pg. 135, finding A-1 with row reductions
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MATLAB Solution of Systems of Linear Equations

• Gaussian Elimination with partial pivoting
• From previous discussion
• det(A) det(L) det(U)
• (A)-1 U-1 L-1
• Ax b can be rewritten as Ax LUx b
• Define z Ux, thus Ax L(Ux) Lz b
• First solve Lz b
• Lastly solve Ux z
• Example 3.16, pg. 136

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so that x 1, 2, 3T
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Gaussian Elimination and Numerical Errors

• cond(A) gives you the condition number of a
  matrix A (how close to singular)
• 1 is well conditioned
• ill-conditioned if a small error in data causes a
  large relative error in the solution
• large condition numbers indicate a an ill
  conditioned matrix
• log10(cond(A))
• Gives you the estimated loss in precision
• Example 3.17, pg. 138

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3.8 Linear Transformations

• An operation is linear if the following is true
• f ?a ?b ? f a ? f b
• Differentiation, integration are linear
• So are transformations such as Laplace and
  Fourier
• Example 3.18, pg. 140
• Theorem 3.9, pg. 141

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Transformations in the Plane
Example pg 143
orthogonal matrix
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3-D rotations
Pg. 144 for rotation matrices Order of rotations
is important, since the matrices do not commute
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Homogeneous Transformations

• The homogenous coordinate representation provides
  for rotation, translation, scaling and
  perspective transformation
• Transformation of an n-D vector in an (n1)-D
  space
• Example 3.19

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Jack B. Kuipers, Quaternions and Rotation
Sequences, Princeton Princeton University
Press, 1999.
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