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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.
Santana Afton
  • 6,758
Samurai Bale
  • 171

3 Answers3

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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

Widawensen
  • 8,172
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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 $A_2$.

Santana Afton
  • 6,758
  • 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 09 '19 at 18:39
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Only one particular case that can be concluded, I think. Since, the determinant is the product of eigenvalues, when every eigenvalue is non-zero then determinant is non-zero. This essentially implies that the matrix equation $AX = 0$ has only trivial solution and therefore nullity is zero. In short:

If every eigenvalue of a characteristic equation is non-zero, then nullity must be zero.

user_1729
  • 75