Stability of linear systems eigenvalues

1. What do eigenvalues tell you about stability?
2. How do you know if a linear system is stable?
3. What are the eigenvalues of a linear operator?
4. What do eigenvalues tell us?

What do eigenvalues tell you about stability?

Eigenvalues can be used to determine whether a fixed point (also known as an equilibrium point) is stable or unstable. A stable fixed point is such that a system can be initially disturbed around its fixed point yet eventually return to its original location and remain there.

How do you know if a linear system is stable?

A standard result in linear algebra tells us that the origin of the system xk+1 = Axk is GAS if and only if all eigenvalues of A have norm strictly less than one; i.e. the spectral radius ρ(A) of A is less than one. In this, we call the matrix A stable (or Schur stable).

What are the eigenvalues of a linear operator?

Definition. Let V be a vector space and L : V → V be a linear operator. A number λ is called an eigenvalue of the operator L if L(v) = λv for a nonzero vector v ∈ V. The vector v is called an eigenvector of L associated with the eigenvalue λ.

What do eigenvalues tell us?

An eigenvalue is a number, telling you how much variance there is in the data in that direction, in the example above the eigenvalue is a number telling us how spread out the data is on the line. The eigenvector with the highest eigenvalue is therefore the principal component.