I was attempting a true/false question from a linear algebra textbook and one of the questions was
"Suppose that the characteristic polynomial of A is p(λ) = λ$^3$ (λ − 2)(λ + 3)$^2$ . Then the nullspace of A can be at most 2–dimensional."
The answer was given as false with the explanation being that the multiplicity of the 0 eigenvalue gives the possible dimensions of the nullspace. In this case, since the multiplicity is 3, the maximum dimension of nullspace can be 3. Hence, the statement is false.
However, I do not understand how the multiplicity of 0 eigenvalue relates to the nullspace. Does that mean that if a characteristic polynomial does not have 0 as its root, the corresponding matrix will have an empty nullspace (besides the 0 vector)? Do the other eigenvalues not have anything to do with the nullspace? What exactly is the relationship between eigenvalues and the nullspace?
Thank you!
Best Answer
By definition of eigenvalues/eigenvectors, we have $Ax=\lambda x$. In the special case of $\lambda = 0$ it becomes $Ax=0$. This means the null space of $A$ is the space that is spanned by the eigenvectors of $0$ eigenvalue.
In your example the algebraic multiplicity of $0$ eigenvalue is 3. However, we don't know the geometric multiplicity - that is the number of linearly independent eigenvectors. But we know that it is at least 1 and cannot exceed the algebraic multiplicity. So the dimension of null space is between 1 and 3 inclusive.