What determines whether or not a matrix is stable?

In summary, the conversation discusses a difficulty in understanding stability and determining whether a matrix is stable. The person posting is seeking a simplified explanation of how to determine stability in terms of the eigenvalues of a matrix. The recommended first step is to compute the determinant of the characteristic equation.
  • #1
Tonyt88
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Basically, I'm having difficulty understanding the concept of stability, I have reread the chapter in my book various times, but to no avail. Can anybody give a very brief overview in simplified terms as to what determines whether or not a matrix is stable. Thanks.
 
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  • #2
You're posting this in the algebra section. So do you mean things like stability conditions of representations of quivers? I doubt it. What topic do you really mean?
 
  • #3
Judging from your earlier post: https://www.physicsforums.com/showthread.php?p=1301361#post1301361 ...

you want to find the stability in terms of the eigenvalues [tex]\lambda[/tex] of an [tex]n\times n[/tex] matrix [tex]A[/tex].

In which case, you first step is to compute the determinant of [tex]\Delta[/tex]; the characteristic equation given by [tex]\Delta=A-\lambda I[/tex].
 

1. What is a stable matrix?

A stable matrix is a matrix that has all its eigenvalues with negative real parts. In other words, for a matrix to be stable, its eigenvalues must be located in the left half of the complex plane.

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

To determine if a matrix is stable, you can calculate its eigenvalues. If all the eigenvalues have negative real parts, then the matrix is stable. Another method is to check if the matrix satisfies the Jury stability criterion, which involves checking certain conditions based on the matrix coefficients.

3. Can a matrix be partially stable?

No, a matrix can only be either stable or unstable. A matrix is considered unstable if at least one of its eigenvalues has a positive real part.

4. What factors affect the stability of a matrix?

The stability of a matrix is determined by its eigenvalues, which in turn are affected by the matrix coefficients. Other factors that can influence stability include the size of the matrix and its initial conditions.

5. Why is it important to determine if a matrix is stable?

Stability is an important concept in many areas of science and engineering, as it indicates the behavior of a system over time. In control systems, for example, a stable matrix ensures that the system will reach a steady state without oscillations or diverging. In other applications, the stability of a matrix can determine the reliability and accuracy of a solution.

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