lcs assignment 03
DESCRIPTION
RughTRANSCRIPT
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Question Apply theorem 7.11 on Exercise 5.2
7.11 Theorem
Given nXn matrix A, if M and Q are symmetric, positive definite, nXn matrices satisfying
QA + ATQ = -M, then all eigenvalues of A have negative real parts. Conversely if all
eigenvalues of A have negative real parts, then for each symmetric nXn matrix M there exist
a unique solution of QA + ATQ = -M given by
Furthermore if M is positive definite, then Q is positive definite.
Exercise 5.2
Solution
Implementing the converse part of the theorem first:
1. If all eigenvalues of A have negative real parts, then for each symmetric
nXn matrix M there exist a unique solution of QA + ATQ = -M given by
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Answer:
Lets generate a random symmetric Matrix M and check whether Q has a unique
solution.
Symmetric matrix would be generated by exploiting the property M+MT is always a
symmetric matrix.
MATLAB Code
syms t A=[0 1; -1 -2]; E=eig(A); if real(E(1,1))
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MATLAB Code:
syms t A=[0 1; -1 -2]; E=eig(A); if real(E(1,1))
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display('Q Matrix is also Positive definite'); else display('Q Matrix is NOT positive definite'); end
else display('All eigenvalues donot have negative real parts ');
end
Output
All eigenvalues have negative real parts
M Matrix is Positive definite
Q =
[ 7/8, 1/2]
[ 1/2, 3/8]
Q Matrix is also Positive definite
Implementing : Given nXn matrix A, if M and Q are symmetric,
positive definite, nXn matrices satisfying QA + ATQ = -M, then
all eigenvalues of A have negative real parts
MATLAB Code
Q=[7/8 1/2; 1/2 3/8]; M=[1 1/2 ; 1/2 1/2]; A=[0 1; -1 -2];
X=(Q*A)+((A.')*Q); Y=-M;
if X==Y display('Equation QA+ATQ=-M holds'); else display('Equation QA+ATQ=-M does not hold'); end
Output
Equation QA+ATQ=-M holds
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The same codes will be used for this question also. Just the values of A and M would
be changed. So outputs would be shown.
Output 1:
All eigenvalues have negative real parts
Q is a Symmetric Matrix
Choose M=[ 2 -1 0]
[-1 2 -1]
[ 0 - 1 2]
Output 2:
All eigenvalues have negative real parts
M Matrix is Positive definite
Q =
[ 3/2, -3/4, 1/2]
[ -3/4, 1, -1/2]
[ 1/2, -1/2, 1]
Q Matrix is also Positive definite
Output 3
Equation QA+ATQ=-M holds
Output1
All eigenvalues have negative real parts
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Q is a Symmetric Matrix
Choose
M =
2 1 1 3
1 2 2 1
1 2 9 1
3 1 1 7
Output2
All eigenvalues have negative real parts
M Matrix is Positive definite
Q =
[ 1, 1/3, 1/2, 7/4]
[ 1/3, 1/2, 2/3, 5/9]
[ 1/2, 2/3, 9/2, 11/4]
[ 7/4, 5/9, 11/4, 25/4]
Q Matrix is also Positive definite
Output 3
Equation QA+ATQ=-M holds