Does the characteristic polynomial encode the rank?

[Bipolarity]

Similar matrices share certain properties, such as the determinant, trace, eigenvalues, and characteristic polynomial. In fact, all of these properties can be determined from the character polynomial alone.

However, similar matrices also share the same rank. I was wondering if the rank is also encoded in the characteristic polynomial of the matrix.

In other words, if two matrices have the same characteristic polynomial, need their rank be the same?

I'd like to know the answer, so that I can decide whether to prove or to cook up a counterexample.

Thanks!

BiP

 

[HallsofIvy]

Well, there is the obvious: the characteristic polynomial of matrix A is $x^rP(x)$ where P is a n- r degree polynomial, if and only if A has rank n- r.

 

[Bipolarity]

Well, there is the obvious: the characteristic polynomial of matrix A is $x^rP(x)$ where P is a n- r degree polynomial, if and only if A has rank n- r.

This seems to be obviously true only for diagonalizable matrices. What if the matrix $A$ is not diagonalizable?

BiP

 

[Office_Shredder]

The algebraic multiplicity of eigenvalues are at least as large as the geometric multiplicities, so if 0 is an eigenvalue with dimension k eigenspace, then the characteristic polynomial has at least a factor of x^k. It's not exactly x^k though, for example the matrix

0 1
0 0

has characteristic polynomial x² but 0 only has a single eigenvector

 

[blue_raver22]

: The characteristic polynomial of a matrix is a unique polynomial that encodes important information about the matrix, such as its eigenvalues and determinant. However, the rank of a matrix is not directly encoded in its characteristic polynomial. While similar matrices share the same characteristic polynomial, they may not necessarily have the same rank. This can be seen in the case of diagonalizable matrices, where the characteristic polynomial is the same but the rank may differ. Therefore, the rank cannot be determined solely from the characteristic polynomial. In order to determine if two matrices have the same rank, we must look at other properties such as row and column operations. It is important to carefully consider all relevant properties when trying to prove or disprove a statement about matrices.