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 \[\begin{align*} \text{det}(xI - M) &= \text{det}(A^{-1} A) \text{det}(xI - M) \\ &= \text{det}(A^{-1}) \text{det}(xI - M) \text{det}(A) \\ &= \text{det}(A^{-1}xIA - A^{-1}MA) \\ &= \text{det}(xA^{-1}A - N) \\ &= \text{det}(xI - N). \end{align*}\] 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