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In linear algebra, 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 field) satisfies its own characteristic equation. If is a given matrix and is the identity matrix, then the characteristic polynomial of is defined as :: where is the determinant operation and λ is a scalar element of the base ring. Since the entries of the matrix are (linear or constant) polynomials in , the determinant is also an -th order monic polynomial in . The Cayley–Hamilton theorem states that substituting the matrix for in this polynomial results in the zero matrix, :: The powers of , obtained by substitution from powers of , are defined by repeated matrix multiplication; the constant term of gives a multiple of the power 0, which power is defined as the identity matrix. The theorem allows to be expressed as a linear combination of the lower matrix powers of . 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 in terms of inverses of linear functions of quaternions, a ''non-commutative'' ring, by Hamilton. This corresponds to the special case of certain real real or complex matrices. The theorem holds for general quaternionic matrices.〔Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see . The rings of quaternions and split-quaternions can both be represented by certain complex matrices. (When restricted to unit norm, these are the groups and respectively.) Therefore it is not surprising that the theorem holds. There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley-Hamilton theorem still holds for the octonions, see .〕 Cayley in 1858 stated it for and smaller matrices, but only published a proof for the case.〔 The general case was first proved by Frobenius in 1878. == Example == As a concrete example, let :. Its characteristic polynomial is given by : The Cayley–Hamilton theorem claims that, if we ''define'' : then : We can verify by computation that indeed, : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cayley–Hamilton theorem」の詳細全文を読む スポンサード リンク
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