## What happens when your eigenvalue is 0?

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.

## What is the eigenvector when the eigenvalue is zero?

Concretely, an eigenvector with eigenvalue 0 is a nonzero vector v such that Av = 0 v , i.e., such that Av = 0. These are exactly the nonzero vectors in the null space of A .

What does an eigenvalue of 0 mean for stability?

Zero Eigenvalues 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.

Can a matrix have eigenvalue zero?

The zero matrix has only zero as its eigenvalues, and the identity matrix has only one as its eigenvalues. In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other.

### Is a matrix Diagonalizable if eigenvalue is 0?

The determinant of a matrix is the product of its eigenvalues. So, if one of the eigenvalues is 0, then the determinant of the matrix is also 0. Hence it is not invertible.

### 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.

Can a square matrix have no eigenvalues?

Over an algebraically closed field, every square matrix has an eigenvalue. For instance, every complex matrix has an eigenvalue. In particular, the existence of eigenvalues for complex matrices is equivalent to the fundamental theorem of algebra. No, but you can build some.

How many Eigen are there?

So the eigenvalues of D are a, b, c, and d, i.e. the entries on the diagonal. This result is valid for any diagonal matrix of any size. So depending on the values you have on the diagonal, you may have one eigenvalue, two eigenvalues, or more.

## How do you find the zero vector?

To find the zero vector, remember that the null vector of a vector space V is a vector 0V such that for all x∈V we have x+0V=x. And this gives a+1=0 and b=0. So the null vector is really (−1,0). The point is: the null vector is defined by properties, axioms, things it must satisfy.

## Is 0 an eigenvalue for an invertible matrix?

No. A matrix is nonsingular (i.e. invertible) iff its determinant is nonzero. and hence, for a nontrivial solution, |λI−A|=0 .

Can you draw a phase portrait with one eigenvalue equal to zero?

Phase portrait with one eigenvalue equal to zero? In my differential equations classes this semester we have been learning how to sketch phase portraits given a solution to a system of equations including eigenvalues and eigenvectors. The cases we have learnt are So you can see they haven’t taught us about zero eigenvalues.

Is the general form of the phase portrait like the above?

But I’d like to know what the general form of the phase portrait would look like in the case that there was a zero eigenvalue. Is the a general case like the above? If so, what is it?

### Is the null space stable if one eigenvalue is zero?

If one eigenvalue is zero than the matrix of the coefficients A has null determinat and a nontrivial null space, and any vector of the null space is an equilibrium point (stable or unstable depending on the sign of the other eigenvalue). See here for some simple example.