Characteristic Polynomial

The characteristic polynomial of a matrix is a polynomial associated to a matrix that gives information about the matrix. It is closely related to the determinant of a matrix, and its roots are the eigenvalues of the matrix. It can be used to find these eigenvalues, prove matrix similarity, or characterize a linear transformation from a vector space to itself.

The characteristic polynomial $p_M(x)$ of an $n \times n$ matrix $M$ is given by $$p_M(x) = \text{det}(xI - M),$$ where $I$ is the identity matrix of rank $n$.

Main Page: Eigenvalues and Eigenvectors

Eigenvalues are the values $\lambda$ satisfying, for some nonzero vector $v$, $M v = \lambda v$.

Let $A$ be an $n$-by-$n$ matrix, so that it corresponds to a transformation $\mathbb{R}^n \to \mathbb{R}^n$. If $\lambda$ is an eigenvalue for $A$, then there is a vector $v \in \mathbb{R}^n$ such that $Av = \lambda v$. Rearranging this equation shows that $(A - \lambda \cdot I)v = 0$, where $I$ denotes the $n$-by-$n$ identity matrix. This implies the null space of the matrix $A-\lambda \cdot I$ is nonzero, so $A-\lambda \cdot I$ has determinant zero.

Note that for every matrix $A$ has 0 as an eigenvalue, with eigenvector $(0,0, \cdots, 0) \in \mathbb{R}^n$. Generally, one is only concerned with the nonzero eigenvectors associated to an eigenvalue, so convention dictates $0$ is considered an eigenvalue of $A$ only when the null space of $A$ is nonzero (equivalently, when $x$ divides $p_{A} (x)$).

Accordingly, any eigenvalue of $A$ must be a root of the polynomial $p_{A} (x) = \det(A - x \cdot I)$. This is called the characteristic polynomial of $A$. Observe that this implies $A$ has only finitely many eigenvalues (in fact, at most $n$ eigenvalues).

In computations, the characteristic polynomial is extremely useful. To determine the eigenvalues of a matrix $A$, one solves for the roots of $p_{A} (x)$, then checks if each root is an eigenvalue.

Two matrices $M$ and $N$ are similar if there exists a matrix $A$ such that $N = A^{-1}MA$. It follows that $$\begin{aligned} \text{det}(xI - M) &= \text{det}(A^{-1} A) \text{det}(xI - M) \\ &= \text{det}(A^{-1}) \text{det}(xI - M) \text{det}(A) \\ &= \text{det}(A^{-1}xIA - A^{-1}MA) \\ &= \text{det}(xA^{-1}A - N) \\ &= \text{det}(xI - N). \end{aligned}$$ So $M$ and $N$ have identical characteristic polynomials.

Combinatorics is, in some sense, the study of the invariants of finite objects. The characteristic polynomial provides a good example of an invariant for certain types of objects, and topics like graphs, posets, and matroids all have associated matrices, the characteristic polynomial of which is of combinatorial importance.

• Eigenvalues and Eigenvectors