Stable matrices and their determinants

In summary, for a dynamical system x(t+1) = Ax(t), the stability of the zero state depends on the determinant of the real n x n matrix A. If |det(A)| > or equal to one, the zero state is stable. If |det(A)| < 1, the zero state is unstable. To understand the stability, one can look at the eigenvalues of A, which can be used as a basis to determine the stability.
  • #1
Tonyt88
62
0

Homework Statement


Consider a dynamical system x(t+1) = Ax(t),, where A is a real n x n matrix.
(a) If |det(A)| > or equal to one, what can you say about the stability of the zero state?
(b) If |det(A)| < 1, what can you say about the stability of the zero state?


Homework Equations





The Attempt at a Solution


I have worked with various matrices knowing if they are stable or not and the value of the determinants, but from what I can see, there exists no relation between the determinant and the stability of the matrix, so basically, where to go from here?
 
Physics news on Phys.org
  • #2
I am used to look at such questions by taking the eigenvectors as basis.
Then everything is understood on the basis of simple numbers: the eigenvalues.
 

1. What is a stable matrix?

A stable matrix is a square matrix that has real eigenvalues with negative real parts. This means that the matrix is stable and will not grow infinitely when multiplied by itself.

2. How is stability of a matrix related to its determinants?

The stability of a matrix is directly related to the sign of its determinant. If a matrix has a positive determinant, it is considered stable, while a negative determinant indicates instability. This is because the determinant represents the scaling factor of the matrix, and a positive scaling factor results in a stable matrix.

3. Can a matrix with a zero determinant be stable?

No, a matrix with a zero determinant is considered unstable. This is because the determinant of a matrix with all zero eigenvalues is equal to zero, indicating that the matrix will not converge to a steady state.

4. How can I determine if a matrix is stable?

To determine if a matrix is stable, you can find the eigenvalues of the matrix and check if they all have negative real parts. If they do, the matrix is stable. Alternatively, you can also check the sign of the determinant, as explained in question 2.

5. What are some applications of stable matrices and their determinants?

Stable matrices and their determinants have various applications in fields such as control systems, economics, physics, and engineering. They are used to model and analyze the stability of systems, such as electrical circuits, chemical reactions, and economic markets. Stable matrices are also used in the design of stable control systems, where the stability of the system is crucial for its proper functioning.

Similar threads

Obtaining stability criterion for 2nd order ODEs in coefficient form
    • Calculus and Beyond Homework Help
Replies
1
Views
253
Proving ##(cof ~A)^t ~A = (det A)I##
    • Calculus and Beyond Homework Help
Replies
4
Views
951
Proof regarding determinant of block matrices
    • Calculus and Beyond Homework Help
Replies
5
Views
4K
Which statements are true given a solution to the system x'=Ax?
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Why is my calculation for the determinant of a matrix incorrect?
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Given specific v, dimension of subspace of L(V, W) where Tv=0?
    • Calculus and Beyond Homework Help
Replies
14
Views
580
Understanding Conditional Expectation, Variance, and Precision Matrices
    • Calculus and Beyond Homework Help
Replies
0
Views
128
Adjacency matrices and network visualisations
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Proof that SU(n) is a group
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Determinant Problem: Show $\det B = \det A$
    • Calculus and Beyond Homework Help
Replies
11
Views
1K

Hot Threads

Recent Insights

Back
Top