Characteristic Polynomial

The characteristic polynomial is a polynomial associated with a square matrix or a linear operator on a finite-dimensional vector space. Defined using a determinant, it records the operator’s eigenvalues and their algebraic multiplicities. In linear algebra, it connects matrix calculations with polynomial factorization and provides a basis-independent description of important properties of a linear operator. Its coefficients include information about the matrix’s trace and determinant, while the Cayley–Hamilton theorem relates the polynomial directly to matrix powers. (math.mit.edu)

Definition and conventions

For an $n\times n$ matrix $A$ with entries in a field $F$, its characteristic polynomial is

where $I_n$ is the identity matrix and $t$ is an indeterminate. Expanding the determinant produces a polynomial with coefficients in $F$, degree $n$, and leading coefficient $1$. A polynomial with leading coefficient $1$ is called monic. The determinant definition also applies to matrices over a commutative ring. (web.mit.edu)

Some texts instead define the characteristic polynomial as $\det(A-tI_n)$. The two conventions differ by the factor $(-1)^n$, so they have identical roots and root multiplicities. The convention $\det(tI_n-A)$ has the advantage of always being monic. Setting either polynomial equal to zero gives the characteristic equation. (math.mit.edu)

For a linear map $T:V\to V$, where $V$ is a finite-dimensional vector space, the characteristic polynomial is defined using a matrix representing $T$ in any basis. Its degree equals the dimension of $V$, and the result is independent of the chosen basis. (ucl.ac.uk)

Eigenvalues and multiplicities

A scalar $\lambda$ is an eigenvalue of $A$ precisely when

Indeed, $Av=\lambda v$ for some nonzero vector $v$ is equivalent to $(\lambda I_n-A)v=0$. Such a vector exists exactly when $\lambda I_n-A$ is singular, or equivalently when its determinant vanishes. Thus eigenvalues can be found by solving a polynomial equation. (math.mit.edu)

Over the complex numbers, the fundamental theorem of algebra ensures that

with roots listed according to multiplicity. The number of times an eigenvalue occurs as a root is its algebraic multiplicity. Its geometric multiplicity is the dimension of the corresponding eigenspace, $\ker(A-\lambda I_n)$, and cannot exceed its algebraic multiplicity. A repeated root therefore does not necessarily imply several linearly independent eigenvectors. (math.mit.edu)

The coefficient field matters. A matrix with real entries can have nonreal eigenvalues, which occur in complex-conjugate pairs. For instance,

has characteristic polynomial $t^2+1$: it has no real eigenvalues, but has eigenvalues $i$ and $-i$ over the complex numbers. (netlib.org)

Coefficients and examples

For $n\geq2$, the characteristic polynomial has the form

The trace is therefore the sum of the eigenvalues, and the determinant their product, in both cases counting algebraic multiplicities. For a $2\times2$ matrix, these quantities determine the entire polynomial; in larger dimensions, additional coefficients are needed. (math.mit.edu)

For example,

gives

Its eigenvalues are $1$ and $3$, consistent with trace $4$ and determinant $3$. More generally, for

(math.mit.edu)

For an upper or lower triangular matrix, the determinant is the product of diagonal entries, yielding

Its eigenvalues are therefore its diagonal entries, including repetitions. (math.mit.edu)

Similarity and diagonalization

Similar matrices have the same characteristic polynomial. If $B=S^{-1}AS$, then

and multiplicativity of determinants gives $p_B(t)=p_A(t)$. This explains the polynomial’s independence from coordinate choices. However, equality of characteristic polynomials does not imply similarity. (textbooks.math.gatech.edu)

For example, $I_2$ and

both have characteristic polynomial $(t-1)^2$, but they are not similar. The first is diagonal, whereas the second has only a one-dimensional eigenspace. More generally, diagonalizability requires a basis of eigenvectors. Having $n$ distinct eigenvalues in the coefficient field guarantees this; repeated eigenvalues require further examination. The characteristic polynomial alone does not describe all the information captured by Jordan normal form. (textbooks.math.gatech.edu)

Cayley–Hamilton theorem and minimal polynomial

The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic polynomial:

Here polynomial evaluation uses matrix powers and replaces the constant term $c$ with $cI_n$. Consequently, $A^n$ is a linear combination of lower powers, and higher powers can be reduced recursively. (web.mit.edu)

The minimal polynomial $m_A(t)$ is the monic polynomial of least degree satisfying $m_A(A)=0$. It divides $p_A(t)$, but need not equal it: for $I_n$, the minimal polynomial is $t-1$, whereas the characteristic polynomial is $(t-1)^n$. The minimal polynomial therefore identifies a potentially shorter polynomial relation obeyed by the matrix. (ucl.ac.uk)

Computation

The determinant formula provides a direct exact calculation, particularly convenient for small or triangular matrices. Numerical eigenvalue software, however, can work directly with matrix transformations rather than explicitly expanding the characteristic polynomial. In numerical linear algebra, the QR algorithm is used to obtain a Schur form, from which eigenvalues are extracted. LAPACK’s nonsymmetric eigenproblem algorithms are normwise backward stable: their computed results correspond to slightly perturbed input matrices, although individual eigenvalues can remain sensitive to perturbations. (netlib.org)