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Part 3Chapter 11
Matrix Inverse and Condition
Part A
All images copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter Objectives
• Know how to determine the matrix inverse in an efficient manner based on LU factorization.
• Understand how the matrix inverse can be used to assess stimulus-response characteristics of engineering systems.
• Understand the meaning of matrix and vector norms and how they are computed.
• Know how to use norms to compute the matrix condition number.
• Understand how the magnitude of the condition number can be used to estimate the precision of solutions of linear algebraic equations.
Matrix Inverse
• If a matrix [A] is square, there is another matrix [A]-1, called the inverse of [A], for which [A][A]-1=[A]-1[A]=[1]
• The inverse can be computed in a column by column fashion by generating solutions with unit vectors as the right-hand-side constants:
A x1 100
A x2
010
A x3
001
A 1 x1 x2 x3 Column vector
Matrix Inverse (cont)
• Recall that LU factorization can be used to efficiently evaluate a system for multiple right-hand-side vectors - thus, it is ideal for evaluating the multiple unit vectors needed to compute the inverse.
Do this once
Do this three times – once for each column
Example 11.1
• Employ LU factorization to determine the matrix inverse for the following system
3 0.1 0.2
0.1 7 0.3
0.3 0.2 10
A
*L d unit
*U x d
In this case x is a column in the inverse
Let’s run this code and watch what happens
Stimulus-Response Computations
• Many systems can be modeled as a linear combination of equations, and thus written as a matrix equation:
• The system response can thus be found using the matrix inverse.
Interactions response stimuli
Interactions response stimuli
A x B
1x A B
1 1 11 11 1 12 2 13 3x a b a b a b
1 1 12 21 1 22 2 23 3x a b a b a b
1 1 13 31 1 32 2 33 3x a b a b a b
Each of the elements represents the response of a single part of the system to a unit stimulus of any other part of the system
Each of these is a dot product
1 1 111 12 13 1
1 1 121 22 23 2
1 1 131 32 33 3
*
a a a b
a a a b
a a a b
11
Suppose there are three jumpers connected by bungee cords, so that each cord is fully extended, but unstretched.
After they are released, gravity takes hold and the jumpers will eventually come to equilibrium
You are asked to compute the displacement of each of the jumpers.
•Gravity causes the stimulus•The displacement is the response
Force Balance
13
21
1 1 2 2 1 1 12
22
2 2 3 3 2 2 1 22
23
3 3 3 2 32
( )
( )
( )
d xm m g k x x k xdt
d xm m g k x x k x x
dt
d xm m g k x xdt
21
1 1 2 2 1 1 12
22
2 2 3 3 2 2 1 22
23
3 3 3 2 32
( ) 0
( ) 0
( ) 0
d xm m g k x x k xdt
d xm m g k x x k x x
dt
d xm m g k x xdt
Since the jumpers are at equilibrium the acceleration is equal to 0
Substituting in the constants we can pose the problem as a system of linear equations
1
2
3
150 100 0 588.6
100 150 50 686.7
0 50 50 784.8
x
x
x
Interactions response stimuli
Force in Newtons (m*g)
Effective Spring Constants
Change in Position
Each element represents the vertical change in position of jumper i, due to a unit change in force applied to jumper j
The position of all three jumpers would change by 0.02 m if the force on jumper 1 was increased by 1 Newton
The position of the first jumper would change by 0.02 m if the force on jumper 2 was increased by 1 Newton – but jumper 2 and 3 would change by 0.03 m
The position of the first jumper would change by 0.02 m if the force on jumper 3 was increased by 1 Newton – jumper 2 would change by 0.03 m and jumper 3 would change by 0.05 m
So why is this useful?
By looking at the matrix inverse I can see where applying a stimulus will have the most (or least) affect.
Error Analysis and System Conditioning
• The inverse also provides a means to see whether systems are ill-conditioned. There are 3 direct methods1. Scale the matrix of coefficients so that the
largest element in each row is 1. Invert the scaled matrix and if there are elements of A-1 that are several orders of magnitude greater than 1 – it is likely that the matrix is ill-conditioned
Error Analysis and System Conditioning
• The inverse also provides a means to see whether systems are ill-conditioned. There are 3 direct methods2. Multiply the inverse by the original matrix, and
see if the result is the identity matrix. If not it is ill-conditioned
Error Analysis and System Conditioning
• The inverse also provides a means to see whether systems are ill-conditioned. There are 3 direct methods3. Invert the inverted matrix and check to see if
the result is equivalent to the original matrix. If not, the system is ill-conditioned
These techniques work, but it would be nice to quantify the problem with a single number
In addition, recall that if the determinant is close to 0, the matrix is ill-conditioned
Vector and Matrix Norms
• A norm is a real-valued function that provides a measure of the size or “length” of multi-component mathematical entities such as vectors and matrices.
• Vector norms and matrix norms may be computed differently.
Vector Norms
• For a vector {X} of size n, the p-norm is:
• Important examples of vector p-norms include:
p1:sum of the absolute values X1 xii1
n
p2 :Euclidian norm (length) X2 X
e xi
2
i1
n
p :maximum magnitude X
1inmax xi
Xp x i
p
i1
n
1/ p
Matrix Norms
• Common matrix norms for a matrix [A] include:
• Note - max is the largest eigenvalue of [A]T[A].
column - sum norm A1
1jnmax aij
i1
n
Frobenius norm Af aij
2
j1
n
i1
n
row - sum norm A
1inmax aij
j1
n
spectral norm (2 norm) A
2 max 1/2
Matrix Condition Number
• The matrix condition number Cond[A] is obtained by calculating Cond[A]=||A||·||A-1||
• In can be shown that:
• The relative error of the norm of the computed solution can be as large as the relative error of the norm of the coefficients of [A] multiplied by the condition number.
• If the coefficients of [A] are known to t digit precision, the solution [X] may be valid to onlyt-log10(Cond[A]) digits.
XX
Cond A AA
MATLAB Commands
• MATLAB has built-in functions to compute both norms and condition numbers:– norm(X,p)
• Compute the p norm of vector X, where p can be any number, inf, or ‘fro’ (for the Euclidean norm)
– norm(A,p)• Compute a norm of matrix A, where p can be 1, 2, inf, or ‘fro’
(for the Frobenius norm)
– cond(X,p) or cond(A,p)• Calculate the condition number of vector X or matrix A using the
norm specified by p.
Example 11.3
• The Hilbert matrix is ‘notoriously’ ill-conditioned.
• Use condition number to help you understand the effect of ill-conditioning on results.