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## Engineering Applications

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Title: Engineering Applications

1
Engineering Applications
• Dr. Darrin Leleux
• Lecture 8 Matrices
• Chapter 3

2
Introduction
• 3.5 Orthogonal and Triangular Matrices
• 3.6 Systems of Linear Equations
• 3.7 MATLAB Matrix Functions
• 3.8 Linear Transformations
• Homework 3

3
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,
4
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

5
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
6
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)

7
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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10
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
11
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

12
Possible states of a system of linear equations
• Consistent, with a unique (one) solution
• Consistent, with infinitely many possible
solutions
• Inconsistent, with no solutions

13
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
14
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

15
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

16
Example 3.13
17
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

18
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

19
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

20
EXAMPLE 3.14
pg 132
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Pg. 135, finding A-1 with row reductions
24
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

25
so that x 1, 2, 3T
26
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

27
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

28
Transformations in the Plane
Example pg 143
orthogonal matrix
29
3-D rotations
Pg. 144 for rotation matrices Order of rotations
is important, since the matrices do not commute
30
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

31
Jack B. Kuipers, Quaternions and Rotation
Sequences, Princeton Princeton University
Press, 1999.
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