page_title: Cayley–Hamilton theorem
section_title: Matrix functions
hierarchical_section_title: Cayley–Hamilton theorem / Applications / Matrix functions
caption_attribution_description: Deutsch: Fotografie des Mathematikers Georg Frobenius English: Photography of the mathematician Georg Frobenius
context_page_description: In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation.
  If A is a given n×n matrix and Iₙ is the n×n identity matrix, then the characteristic polynomial of A is defined as
  where det is the determinant operation and λ is a variable for a scalar element of the base ring. Since the entries of the matrix are polynomials in λ, the determinant is also an n-th order monic polynomial in λ. The Cayley–Hamilton theorem states that if one defines an analogous matrix equation, p, consisting of the replacement of the scalar variable λ with the matrix A, then this polynomial in the matrix A results in the zero matrix,
  The powers of A, obtained by substitution from powers of λ, are defined by repeated matrix multiplication; the constant term of p gives a multiple of the power A⁰, which is defined as the identity matrix.
  The theorem allows Aⁿ to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.
context_section_description: Given an analytic function 
  and the characteristic polynomial p(x)  of degree n of an n × n matrix A, the function can be expressed using long division as 
  where q(x) is some quotient polynomial and  r(x) is a remainder polynomial such that 0 ≤ deg r(x) < n.
  By the Cayley–Hamilton theorem, replacing x by the matrix A gives p(A) = 0, so one has
  Thus, the analytic function of the matrix A can be expressed as a matrix polynomial of degree less than n.
  Let the remainder polynomial be
  Since p(λ) = 0, evaluating the function f(x) at the n eigenvalues of A,  yields
  This amounts to a system of n linear equations, which can be solved to determine the coefficients cᵢ. Thus, one has 
  When the eigenvalues are repeated, that is λᵢ = λⱼ for some i ≠ j, two or more equations are identical; and hence the linear equations cannot be solved uniquely. For such cases, for an eigenvalue λ with multiplicity m, the first m – 1 derivatives of p(x) vanish at the eigenvalue. This 
  leads to the extra m – 1 linearly independent solutions 
  which,   combined with others, yield the required n equations to solve for cᵢ.
  Finding a polynomial that passes through the points (λᵢ, f (λᵢ)) is essentially an interpolation problem, and can be solved using Lagrange or Newton interpolation techniques,  leading to  Sylvester's formula.
  For example, suppose the task is to find the polynomial representation of 
  The characteristic polynomial is p(x) = (x − 1)(x − 3) = x² − 4x + 3, and the eigenvalues are λ = 1, 3. Let r(x) = c₀ + c₁x. Evaluating f(λ) = r(λ) at the eigenvalues, one obtains two linear equations, eᵗ = c₀ + c₁ and e³ᵗ = c₀ + 3c₁.
  Solving the equations yields c₀ = (3eᵗ − e³ᵗ)/2 and c₁ = (e³ᵗ − eᵗ)/2. Thus, it follows that
  If, instead, the function were f(A) = sin At, then the coefficients would have been c₀ = (3 sin t − sin 3t)/2 and c₁ = (sin 3t - sin t)/2; hence
  As a further example, when considering
  then the characteristic polynomial is p(x) = x² + 1, and the eigenvalues are λ = ±i.
  As before, evaluating the function at the eigenvalues gives us the linear equations eⁱᵗ = c₀ + i c₁ and e⁻ⁱᵗ = c₀ − ic₁; the solution of which gives, c₀ = (eⁱᵗ + e⁻ⁱᵗ)/2 = cos t and c₁ = (eⁱᵗ − e⁻ⁱᵗ)/2i = sin t. Thus, for this case,
  which is a rotation matrix.
  Standard examples of such usage is the exponential map from the Lie algebra of a matrix Lie group into the group. It is given by a matrix exponential,
  Such expressions have long been known for SU(2),
  where the σ are the Pauli matrices and for SO(3),
  which is Rodrigues' rotation formula. For the notation, see rotation group SO(3)#A note on Lie algebra.
  More recently, expressions have appeared for other groups, like the Lorentz group SO(3, 1), O(4, 2) and SU(2, 2), as well as GL(n, R). The group O(4, 2) is the conformal group of spacetime, SU(2, 2) its simply connected cover (to be precise, the simply connected cover of the connected component SO⁺(4, 2) of O(4, 2)). The expressions obtained apply to the standard representation of these groups. They require knowledge of (some of) the eigenvalues of the matrix to exponentiate. For SU(2) (and hence for SO(3)), closed expressions have   been obtained for all irreducible representations, i.e. of any spin.
caption: Ferdinand Georg Frobenius (1849–1917), German mathematician. His main interests were elliptic functions, differential equations, and later group theory.
  In 1878 he gave the first full proof of the Cayley–Hamilton theorem.[10]
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