What does it mean if 0 is an eigenvalue?
A zero eigenvalue means the matrix in question is singular. The eigenvectors corresponding to the zero eigenvalues form the basis for the null space of the matrix.
What happens when both eigenvalues are zero?
If the eigenvalue A equals 0 then Ax = 0x = 0. Vectors with eigenvalue 0 make up the nullspace of A; if A is singular, then A = 0 is an eigenvalue of A. Suppose P is the matrix of a projection onto a plane.
Can eigen values be 0?
Eigenvalues may be equal to zero. We do not consider the zero vector to be an eigenvector: since A 0 = 0 = λ 0 for every scalar λ , the associated eigenvalue would be undefined.
What does a zero eigenvalue mean for stability?
If an eigenvalue has no imaginary part and is equal to zero, the system will be unstable, since, as mentioned earlier, a system will not be stable if its eigenvalues have any non-negative real parts. This is just a trivial case of the complex eigenvalue that has a zero part.
Can zero be an eigenvalue for a square matrix?
We do not consider the zero vector to be an eigenvector: since A 0 = 0 = λ 0 for every scalar λ , the associated eigenvalue would be undefined. Yes. 0 is an eigenvalue of a square matrix A if and only if there is a nonzero vector v with Av=0.
Is the zero solution stable unstable or asymptotically stable?
In the first case, λ = 3 is an eigenvalue, so the zero solution is unstable. In the second case, the eigenvalues are λ = ±i, so the zero solution is stable but not asymptotically stable. In the third case, both eigenvalues are negative, so the zero solution is (asymptotically) stable.
How do you tell if a differential equation is stable or unstable?
If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. If a solution does not have either of these properties, it is called unstable.
Is zero a distinct eigenvalue?
Yes the eigenvalues are 0 1 and 2 but is 0 distinct? Yes, it is no more special than having 1 or 2. All it means is that your matrix is singular.