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Engineering Applications

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

Introduction

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

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

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

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)

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

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

Possible states of a system of linear equations

- Consistent, with a unique (one) solution
- Consistent, with infinitely many possible

solutions - Inconsistent, with no solutions

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

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

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

Example 3.13

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

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

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

EXAMPLE 3.14

pg 132

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Pg. 135, finding A-1 with row reductions

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

so that x 1, 2, 3T

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

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

Transformations in the Plane

Example pg 143

orthogonal matrix

3-D rotations

Pg. 144 for rotation matrices Order of rotations

is important, since the matrices do not commute

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

Jack B. Kuipers, Quaternions and Rotation

Sequences, Princeton Princeton University

Press, 1999.

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