# Given a characteristic polynomial of a matrix A, find the nullity, rank

If i'm only given the characteristic polynomial of a matrix $$A$$, how would I find the given information:

• Rank
• Nullity
• Number of elements in the eigenvectors of $$A$$
• Number of elements in the null space of $$A$$

The only information that i can figure out are:

• the eigenvalues
• the dimensions of the matrix and
• the determinant of the matrix which is the product of the eigenvalues.

## 2 Answers

In general case it's impossible to know the rank of a matrix only from its characteristic polynomial

see the answer in Relation between rank and number of distinct eigen values

If you only have the characteristic polynomial, then you can’t find much about the rank or nullity of $$A$$. Consider the following examples:

$$A_1=\begin{bmatrix} 0 & 1\\0&0\end{bmatrix};\qquad A_2 = \begin{bmatrix}0&0\\0&0\end{bmatrix}.$$

The characteristic polynomial of both matrices is $$x^2$$, but the rank and nullity of $$A_1$$ is different from that of $$A_2$$.

• The most information you can gather about rank or nullity from the characteristic polynomial is that nullity is at least $1$ if $x$ is a factor, and the rank is at least $k$ if there are $k$ distinct linear factors not equal to $x$. – Santana Afton Mar 9 at 18:39