1 Statement of the Cayley–Hamilton theorem

The Cayley–Hamilton theorem asserts that a square matrix satisfies the algebraic equation given by its characteristic polynomial. This provides a universal polynomial identity that holds regardless of diagonalizability or other special structure.

1.1 Characteristic polynomial of a matrix

For an \(n\times n\) matrix \(A\) over a commutative ring (or field) \(R\), the characteristic polynomial is defined by \[ p_A(\lambda)=\det(\lambda I-A). \] It is a monic polynomial in \(\lambda\) of degree \(n\).

1.2 Matrix substitution into a polynomial

Given a polynomial \(p(\lambda)=c_0+c_1\lambda+\cdots+c_n\lambda^n\), the expression \(p(A)\) means the matrix obtained by replacing \(\lambda^k\) with \(A^k\), i.e. \[ p(A)=c_0I+c_1A+\cdots+c_nA^n. \]

1.3 Formal theorem statement and interpretation

If \(A\) is an \(n\times n\) matrix and \(p_A(\lambda)\) is its characteristic polynomial, then \[ p_A(A)=0, \] where \(0\) denotes the zero matrix of size \(n\times n\). Conceptually, the theorem says that the characteristic polynomial encodes a polynomial relation among the powers of \(A\).

1.4 Scope of validity (rings vs. fields)

The standard formulation requires coefficients to lie in a commutative ring so that determinants and polynomial coefficients behave compatibly with matrix entries. Over fields, the theorem is particularly common in applications to eigenvalues and spectral theory, but the identity itself is fundamentally algebraic and extends to suitable commutative coefficient systems.

2 Examples and concrete calculations

Working through low-dimensional cases illustrates how the characteristic polynomial produces explicit matrix equations.

2.1 \(2\times 2\) matrices

Consider \(A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\).

2.1.1 Computing the characteristic polynomial

One finds \[ p_A(\lambda)=\det(\lambda I-A)=\det\begin{pmatrix}\lambda-a&-b\\ -c&\lambda-d\end{pmatrix} =(\lambda-a)(\lambda-d)-bc, \] so \[ p_A(\lambda)=\lambda^2-(a+d)\lambda+(ad-bc). \]

2.1.2 Verifying \(p_A(A)=0\)

Cayley–Hamilton gives \[ A^2-(\operatorname{tr}A)A+(\det A)I=0. \] Substituting \(A\) into the polynomial yields a concrete matrix identity; direct multiplication confirms that the left-hand side becomes the zero matrix.

2.2 \(3\times 3\) matrices

For \(A\in M_3(R)\), the characteristic polynomial takes the form \[ p_A(\lambda)=\lambda^3-c_1\lambda^2+c_2\lambda-c_3 \] for suitable coefficients \(c_1,c_2,c_3\) determined by traces of exterior powers (or equivalently by minors). Cayley–Hamilton then yields a cubic matrix relation: \[ A^3-c_1A^2+c_2A-c_3I=0. \] While the explicit coefficients can be computed from determinants of \(\lambda I-A\), the conceptual point is that higher powers of \(A\) can always be rewritten in terms of \(I,A,A^2\).

2.3 Special cases (diagonal, triangular, nilpotent matrices)

  • Diagonal matrices: If \(A=\mathrm{diag}(\alpha_1,\dots,\alpha_n)\), then \(p_A(\lambda)=\prod_{i=1}^n(\lambda-\alpha_i)\). Substituting \(A\) makes each diagonal entry satisfy the corresponding scalar polynomial, producing \(p_A(A)=0\) entrywise.
  • Triangular matrices: For an upper (or lower) triangular matrix, the eigenvalues are the diagonal entries, and the characteristic polynomial is the product \(\prod (\lambda-\text{diag entry})\). Cayley–Hamilton still gives a full matrix identity, not merely a diagonal one.
  • Nilpotent matrices: If \(A^k=0\) for some \(k\), then the characteristic polynomial must have \(\lambda\) as a factor; indeed \(p_A(0)=\det(-A)=0\). Cayley–Hamilton then reduces to a polynomial relation compatible with nilpotency.

3 Relationship to polynomial identities

Cayley–Hamilton is an instance of a more general phenomenon: matrices satisfy polynomial equations coming from their algebraic invariants.

3.1 Minimal polynomial and its connection

The minimal polynomial \(m_A(\lambda)\) of \(A\) is the monic polynomial of least degree such that \(m_A(A)=0\). By definition, \(m_A\) divides any polynomial that annihilates \(A\), including the characteristic polynomial. Thus, \[ m_A(\lambda)\mid p_A(\lambda). \]

3.2 When the characteristic polynomial equals the minimal polynomial

The two polynomials coincide when there is no nontrivial reduction of degree needed to annihilate \(A\). This happens, for example, when the size and structure of Jordan blocks (over an algebraic closure, when relevant) force the annihilating relation to require the full degree \(n\). In such cases, the characteristic polynomial provides the “tightest” polynomial identity for \(A\).

3.3 Consequences for commuting matrices and polynomial functions

If matrices commute, polynomial relations become especially useful. When \(A\) commutes with \(B\), polynomials in \(A\) and \(B\) can often be manipulated simultaneously. Cayley–Hamilton then supplies reduction rules that express high-degree polynomial functions of a matrix in terms of lower-degree ones, making it possible to simplify expressions like \(f(A)\) for rational functions \(f\) defined through polynomial identities.

4 Applications

The theorem’s primary utility is that it turns high-degree matrix expressions into lower-degree ones, enabling practical computations and theoretical bounds.

4.1 Computing matrix powers using characteristic polynomials

Since \(p_A(A)=0\), any power \(A^k\) with \(k\ge n\) can be reduced to a linear combination of lower powers \(I,A,\dots,A^{n-1}\). For example, in the \(2\times 2\) case the identity \[ A^2=(\operatorname{tr}A)A-(\det A)I \] immediately yields formulas for \(A^3,A^4,\dots\) by successive substitution.

4.2 Finding inverses via polynomial expressions

If \(A\) is invertible, Cayley–Hamilton can be rearranged to produce an expression for \(A^{-1}\) as a polynomial in \(A\). For instance, from a degree-\(n\) relation \[ A^n+c_{n-1}A^{n-1}+\cdots+c_1A+c_0I=0, \] multiplying by \(A^{-1}\) (and using \(c_0\neq 0\) equivalent to \(\det A\neq 0\)) yields \[ A^{-1}=\text{(a polynomial in }A\text{ of degree }n-1). \]

4.3 Deriving reduction formulas for rational functions of matrices

Rational expressions \(r(A)\) can often be interpreted through polynomials by clearing denominators and using Cayley–Hamilton to reduce degrees. When a rational function is defined on the spectrum of \(A\), its evaluation can be rewritten as a polynomial in \(A\) of bounded degree. This underlies efficient computation of matrix functions such as resolvents \((A-\lambda I)^{-1}\) in theoretical settings.

4.4 Dimension bounds for spans of \(\{I,A,A^2,\dots\}\)

Cayley–Hamilton implies that the set of matrices \[ \{I,A,A^2,\dots\} \] spans a space of dimension at most \(n\), since \(A^n\) and higher powers lie in the span of \(\{I,A,\dots,A^{n-1}\}\). More precisely, the dimension equals the degree of the minimal polynomial.

5 Eigenvalues and spectral consequences

While Cayley–Hamilton is stated as a matrix identity, it tightly constrains eigen-structure when considered over suitable coefficient extensions.

5.1 Eigenvalues as roots of the characteristic polynomial

If \(A\) is over a field and \(\alpha\) is an eigenvalue with eigenvector \(v\neq 0\), then \[ (\alpha I-A)v=0 \quad\Rightarrow\quad \det(\alpha I-A)=0. \] Hence \(\alpha\) must be a root of \(p_A(\lambda)\). Cayley–Hamilton then reinforces the idea that the characteristic polynomial captures all eigenvalues (counted with algebraic multiplicity).

5.2 Invariant subspaces and polynomial annihilation

For an eigenvalue \(\alpha\), the space of vectors annihilated by \((A-\alpha I)^k\) for varying \(k\) forms the chain of generalized eigenspaces. Cayley–Hamilton implies that applying \(p_A(A)\) annihilates every vector, which means the entire space decomposes into pieces controlled by these polynomial factors. Polynomial annihilation therefore organizes invariant subspaces tied to the roots of \(p_A\).

5.3 Jordan form perspective (conceptual)

Over an algebraically closed field, \(A\) can be expressed (up to similarity) in Jordan canonical form. The theorem corresponds, on each Jordan block, to the statement that the block satisfies the same characteristic polynomial equation. This provides a conceptual bridge: the structure of Jordan blocks determines how the characteristic polynomial factors, and Cayley–Hamilton forces the resulting polynomial relation on each block simultaneously.

6 Proofs of the theorem

Several proof strategies exist; they differ in how the characteristic polynomial is connected to matrix algebra.

6.1 Proof via adjugate matrix and determinant identities

A common route uses the adjugate matrix \(\operatorname{adj}(X)\), defined by the identity \[ X\,\operatorname{adj}(X)=\det(X)I. \] Apply this with \(X=\lambda I-A\). Then \[ (\lambda I-A)\operatorname{adj}(\lambda I-A)=p_A(\lambda)I. \] Expanding the adjugate as a matrix whose entries are polynomials in \(\lambda\) and comparing coefficients allows one to derive that substituting \(A\) into the characteristic polynomial annihilates \(A\), yielding \(p_A(A)=0\).

6.2 Proof using companion matrices

Another proof uses the fact that every monic polynomial \(p(\lambda)\) of degree \(n\) is the characteristic polynomial of its companion matrix. For a given matrix \(A\), one can relate the action of \(A\) to multiplication by \(\lambda\) in the quotient module \(R[\lambda]/(p_A(\lambda))\). Under this correspondence, Cayley–Hamilton becomes the statement that multiplication by \(\lambda\) on the quotient satisfies the defining relation, forcing \(p_A(A)=0\).

6.3 Proof sketch through polynomial division and invariance

One can show that for any polynomial \(q(\lambda)\), the matrices \(q(A)\) can be reduced modulo the characteristic polynomial to obtain a remainder of degree \(<n\). By analyzing how the remainder behaves under multiplication by \(A\) and using linear dependence among \(\{I,A,\dots,A^n\}\), it follows that the specific polynomial \(p_A\) yields the zero remainder when evaluated at \(A\). This perspective emphasizes reduction to a canonical basis of powers.

6.4 Proof in terms of the regular representation (conceptual overview)

From an abstract algebra viewpoint, the theorem can be phrased using the regular representation of the matrix’s action on a module generated by powers of \(A\). The characteristic polynomial arises as the determinant of \(\lambda I\) minus this linear transformation, and the Cayley–Hamilton identity reflects that the transformation satisfies the polynomial determined by its characteristic data. This approach highlights that the theorem is ultimately about linear endomorphisms and determinants.

Cayley–Hamilton has many adaptations depending on coefficient systems, functional interpretation, and block structures.

7.1 Cayley–Hamilton over different coefficient systems

The identity holds over commutative rings where determinants are defined and polynomial substitution remains meaningful. Over fields, it is routinely used with eigenvalues and algebraic multiplicities. Over general commutative rings, the statement remains valid but can require careful interpretation of determinants and factors.

7.2 Matrix polynomial functional calculus viewpoint

Cayley–Hamilton implies that evaluating polynomials at \(A\) depends only on the residue class of the polynomial modulo \(p_A(\lambda)\). Consequently, the “polynomial calculus” for \(A\) collapses to a finite-dimensional quotient: \[ R[\lambda]/(p_A(\lambda)). \] This viewpoint explains why rational functions (when defined) can often be reduced to polynomial expressions of bounded degree.

7.3 Relation to Newton identities and trace power sums

Coefficients of the characteristic polynomial can be expressed in terms of traces of powers of \(A\) (or traces on exterior powers). Newton identities provide the bridge between these quantities: they relate elementary symmetric polynomials (appearing in \(p_A\)) to power sums \(\operatorname{tr}(A^k)\). Through Cayley–Hamilton, such trace relations connect directly to polynomial annihilation.

7.4 Generalizations to polynomial matrices and block structures

There are generalizations where one studies polynomial matrices or matrices with block decompositions, using characteristic polynomials or determinantal identities for larger algebraic objects. While the specific statement changes with the generalized setting, the guiding idea persists: determinants and characteristic data yield polynomial constraints that reduce computations and describe algebraic dependence.

8 Computational considerations

While the theorem is exact and symbolic in nature, practical computation raises separate issues.

8.1 Numerical stability vs. symbolic verification

In floating-point arithmetic, evaluating \(p_A(A)\) directly can be numerically unstable due to cancellation and growth of intermediate powers. For symbolic verification, by contrast, the theorem is straightforward to apply once the characteristic polynomial is known. In numerical linear algebra, one therefore typically uses the theorem for algebraic simplification or for exact arithmetic, rather than naive direct evaluation.

8.2 Efficient evaluation of \(p_A(A)\)

When coefficients are available, the most efficient computation of \(p_A(A)\) uses degree reduction: compute \(A^k\) only up to \(k=n-1\), and express higher powers via the Cayley–Hamilton relation. Horner-like schemes for polynomial evaluation can also reduce the number of matrix multiplications when implementing polynomial functions of \(A\).

8.3 Using the theorem to reduce algorithmic complexity

The theorem can lower the computational burden of tasks involving high powers, repeated multiplication, or polynomial/rational functions of matrices. By converting expressions into a bounded-degree basis \(\{I,A,\dots,A^{n-1}\}\), one reduces both time and memory requirements. The payoff is greatest in algorithms that repeatedly evaluate or manipulate matrix functions, where degree truncation prevents growth in computational complexity.