# Cayley–Hamilton theorem

Cayley–Hamilton theorem Arthur Cayley, F.R.S. (1821–1895) is widely regarded as Britain's leading pure mathematician of the 19th century. Cayley in 1848 went to Dublin to attend lectures on quaternions by Hamilton, their discoverer. Later Cayley impressed him by being the second to publish work on them.[1] Cayley proved the theorem for matrices of dimension 3 and less, publishing proof for the two-dimensional case.[2][3] As for n × n matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”. William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathematician, first foreign member of the American National Academy of Sciences. While maintaining opposing position about how geometry should be studied, Hamilton always remained on the best terms with Cayley.[1] Hamilton proved that for a linear function of quaternions there exists a certain equation, depending on the linear function, that is satisfied by the linear function itself.[4][5][6] Em álgebra linear, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) 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[7] {estilo de exibição p_{UMA}(lambda )= isso(lambda I_{n}-UMA)} , 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 {estilo de exibição (lambda I_{n}-UMA)} são (linear or constant) polynomials in λ, the determinant is also a degree-n monic polynomial in λ, {estilo de exibição p_{UMA}(lambda )=lambda ^{n}+c_{n-1}lambda ^{n-1}+cdots +c_{1}lambda +c_{0}~.} One can create an analogous polynomial {estilo de exibição p_{UMA}(UMA)} in the matrix A instead of the scalar variable λ, definido como {estilo de exibição p_{UMA}(UMA)=A^{n}+c_{n-1}A^{n-1}+cdots +c_{1}A+c_{0}EU_{n}~.} The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that {estilo de exibição p_{UMA}(UMA)= mathbf {0} } . 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. The theorem was first proved in 1853[8] in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton.[4][5][6] This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices.[9][nb 1] Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.[2] The general case was first proved by Ferdinand Frobenius in 1878.[10] Conteúdo 1 Exemplos 1.1 1 × 1 matrices 1.2 2 × 2 matrices 2 Formulários 2.1 Determinant and inverse matrix 2.2 n-th power of matrix 2.3 Matrix functions 2.4 Algebraic number theory 3 Provas 3.1 Preliminaries 3.1.1 Adjugate matrices 3.2 A direct algebraic proof 3.3 A proof using polynomials with matrix coefficients 3.4 A synthesis of the first two proofs 3.5 A proof using matrices of endomorphisms 3.6 A bogus "proof": p(UMA) = det(AIn − A) = det(A − A) = 0 3.7 Proofs using methods of abstract algebra 4 Abstraction and generalizations 5 Veja também 6 Observações 7 Notas 8 Referências 9 External links Examples 1 × 1 matrices For a 1 × 1 matrix A = (uma), the characteristic polynomial is given by p(λ) = λ − a, and so p(UMA) = (uma) − a(1) = 0 é trivial.

2 × 2 matrices As a concrete example, deixar {estilo de exibição A={começar{pmatrix}1&2\3&4end{pmatrix}}.} Its characteristic polynomial is given by {estilo de exibição p(lambda )= isso(lambda I_{2}-UMA)= isso !{começar{pmatrix}lambda -1&-2\-3&lambda -4end{pmatrix}}=(lambda -1)(lambda -4)-(-2)(-3)=lambda ^{2}-5lambda -2.} The Cayley–Hamilton theorem claims that, if we define {estilo de exibição p(X)=X^{2}-5X-2I_{2},} então {estilo de exibição p(UMA)=A^{2}-5A-2I_{2}={começar{pmatrix}0&0\0&0\end{pmatrix}}.} We can verify by computation that indeed, {estilo de exibição A^{2}-5A-2I_{2}={começar{pmatrix}7&10\15&22\end{pmatrix}}-{começar{pmatrix}5&10\15&20\end{pmatrix}}-{começar{pmatrix}2&0\0&2\end{pmatrix}}={começar{pmatrix}0&0\0&0\end{pmatrix}}.} For a generic 2 × 2 matrix, {estilo de exibição A={começar{pmatrix}a&b\c&d\end{pmatrix}},} the characteristic polynomial is given by p(λ) = λ2 − (uma + d)λ + (ad − bc), so the Cayley–Hamilton theorem states that {estilo de exibição p(UMA)=A^{2}-(a+d)A+(ad-bc)EU_{2}={começar{pmatrix}0&0\0&0\end{pmatrix}};} which is indeed always the case, evident by working out the entries of A2.

show Proof Applications Determinant and inverse matrix See also: Determinant § Relation to eigenvalues and trace, and Characteristic polynomial § Properties For a general n × n invertible matrix A, ou seja, one with nonzero determinant, A−1 can thus be written as an (n − 1)-th order polynomial expression in A: As indicated, the Cayley–Hamilton theorem amounts to the identity {estilo de exibição p(UMA)=A^{n}+c_{n-1}A^{n-1}+cdots +c_{1}A+(-1)^{n}a(UMA)EU_{n}=0.} The coefficients ci are given by the elementary symmetric polynomials of the eigenvalues of A. Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues: {estilo de exibição s_{k}=soma _{i=1}^{n}lambda _{eu}^{k}=nome do operador {tr} (A^{k}),} where tr(Ak) is the trace of the matrix Ak. Desta forma, we can express ci in terms of the trace of powers of A.

No geral, the formula for the coefficients ci is given in terms of complete exponential Bell polynomials as[nb 2] {estilo de exibição c_{n-k}={fratura {(-1)^{k}}{k!}}B_{k}(s_{1},-1!s_{2},2!s_{3},ldots ,(-1)^{k-1}(k-1)!s_{k}).} Em particular, the determinant of A equals (−1)nc0. Desta forma, the determinant can be written as the trace identity: {exibi-lo(UMA)={fratura {1}{n!}}B_{n}(s_{1},-1!s_{2},2!s_{3},ldots ,(-1)^{n-1}(n-1)!s_{n}).} Da mesma maneira, the characteristic polynomial can be written as {estilo de exibição -(-1)^{n}a(UMA)EU_{n}=A(A^{n-1}+c_{n-1}A^{n-2}+cdots +c_{1}EU_{n}),} e, by multiplying both sides by A−1 (note −(−1)n = (−1)n−1), one is led to an expression for the inverse of A as a trace identity, {estilo de exibição {começar{alinhado}A^{-1}&={fratura {(-1)^{n-1}}{det A}}(A^{n-1}+c_{n-1}A^{n-2}+cdots +c_{1}EU_{n}),\[5pt]&={fratura {1}{det A}}soma _{k=0}^{n-1}(-1)^{n+k-1}{fratura {A^{n-k-1}}{k!}}B_{k}(s_{1},-1!s_{2},2!s_{3},ldots ,(-1)^{k-1}(k-1)!s_{k}).fim{alinhado}}} Another method for obtaining these coefficients ck for a general n × n matrix, provided no root be zero, relies on the following alternative expression for the determinant, {estilo de exibição p(lambda )= isso(lambda I_{n}-UMA)=lambda ^{n}exp(nome do operador {tr} (registro(EU_{n}-A/lambda ))).} Por isso, by virtue of the Mercator series, {estilo de exibição p(lambda )=lambda ^{n}exp esquerda(-nome do operador {tr} soma _{m=1}^{infty }{({A over lambda })^{m} over m}certo),} where the exponential only needs be expanded to order λ−n, since p(λ) is of order n, the net negative powers of λ automatically vanishing by the C–H theorem. (Novamente, this requires a ring containing the rational numbers.) Differentiation of this expression with respect to λ allows one to express the coefficients of the characteristic polynomial for general n as determinants of m × m matrices,[nb 3] {estilo de exibição c_{n-m}={fratura {(-1)^{m}}{m!}}{começar{vmatrix}nome do operador {tr} A&m-1&0&cdots \operatorname {tr} A^{2}&operatorname {tr} A&m-2&cdots \vdots &vdots &&&vdots \operatorname {tr} A^{m-1}&operatorname {tr} A^{m-2}&cdots &cdots &1\operatorname {tr} A^{m}&operatorname {tr} A^{m-1}&cdots &cdots &operatorname {tr} Aend{vmatrix}}~.} Examples For instance, the first few Bell polynomials are B0 = 1, B1(x1) = x1, B2(x1, x2) = x2 1 + x2, and B3(x1, x2, x3) = x3 1 + 3 x1x2 + x3.