Editing Cayley–Hamilton theorem (section)

{{short description|Every square matrix over a commutative ring satisfies its own characteristic equation}}
[[File:Arthur Cayley.jpg|225px|thumb|right|[[Arthur Cayley]], [[Fellow of the Royal Society|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 [[quaternion]]s by Hamilton, their discoverer. Later Cayley impressed him by being the second to publish work on them.<ref name=Crilly_1>{{harvnb|Crilly|1998}}</ref>
Cayley stated the theorem for matrices of dimension 3 or less, and published a proof for the two-dimensional case.]]
[[File:William Rowan Hamilton portrait oval combined.png|225px|thumb|right|[[William Rowan Hamilton]] (1805–1865), Irish physicist, astronomer, and mathematician, first foreign member of the American [[National Academy of Sciences]]. While maintaining an opposing position about how [[geometry]] should be studied, Hamilton always remained on the best terms with Cayley.<ref name=Crilly_1/><br/><br/>Hamilton proved that for a linear function of [[quaternion]]s there exists a certain equation, depending on the linear function, that is satisfied by the linear function itself.<ref name=Hamilton_1864a/><ref name=Hamilton_1864b/><ref name=Hamilton_1862/>]]

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 number|real]] or [[complex number]]s or the [[Integer#Algebraic_properties|integers]]) satisfies its own [[Characteristic polynomial#Characteristic equation|characteristic equation]].

The [[characteristic polynomial]] of an {{math|''n''&thinsp;×&thinsp;''n''}} matrix {{mvar|A}} is defined as<ref>{{harvnb|Atiyah|MacDonald|1969}}</ref> <math>p_A(\lambda)=\det(\lambda I_n-A)</math>, where {{math|det}} is the [[determinant]] operation, {{mvar|λ}} is a [[variable (mathematics)|variable]] [[scalar (mathematics)|scalar]] element of the base [[ring (mathematics)|ring]], and {{math|''I<sub>n</sub>''}} is the {{math|''n''&thinsp;×&thinsp;''n''}} [[identity matrix]]. Since each entry of the matrix <math>(\lambda I_n-A)</math> is either constant or linear in {{mvar|λ}}, the determinant of <math>(\lambda I_n-A)</math> is a [[degree of a polynomial|degree]]-{{mvar|n}} [[monic polynomial|monic]] [[polynomial]] in {{mvar|λ}}, so it can be written as 
<math display="block">p_A(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \cdots + c_1\lambda + c_0.</math> By replacing the scalar variable {{Mvar|λ}} with the matrix {{Mvar|A}}, one can define an analogous [[matrix  polynomial]] expression,  <math display="block">p_A(A) = A^n + c_{n-1}A^{n-1} + \cdots + c_1A + c_0I_n.</math>
(Here, <math>A</math> is the given matrix—not a variable, unlike <math>\lambda</math>—so <math>p_A(A)</math> is a constant rather than a function.)
The Cayley–Hamilton theorem states that this polynomial expression is equal to the [[zero matrix]], which is to say that <math>p_A(A) = \mathbf 0;</math> that is, the characteristic polynomial <math>p_A</math> is an [[annihilating polynomial]] for <math>A.</math>  

One use for the Cayley–Hamilton [[theorem]] is that it  allows {{mvar|A}}<sup>{{mvar|n}}</sup> to be expressed as a [[linear combination]] of the lower matrix powers of {{mvar|A}}: 
<math display="block">A^n = -c_{n-1}A^{n-1} - \cdots - c_1A - c_0I_n.</math>
When the ring is a [[field (mathematics)|field]], the Cayley–Hamilton theorem is equivalent to the statement that the [[Minimal polynomial (linear algebra)|minimal polynomial]] of a square matrix [[Polynomial division|divides]] its characteristic polynomial.

A special case of the theorem was first proved by Hamilton in 1853<ref name=Hamilton_1853>{{harvnb|Hamilton|1853|p=562}}</ref> in terms of inverses of linear functions of [[quaternion]]s.<ref name=Hamilton_1864a>{{harvnb|Hamilton|1864a}}</ref><ref name=Hamilton_1864b>{{harvnb|Hamilton|1864b}}</ref><ref name=Hamilton_1862>{{harvnb|Hamilton|1862}}</ref> This corresponds to the special case of certain {{math|4&thinsp;×&thinsp;4}} real or {{math|2&thinsp;×&thinsp;2}} complex matrices. Cayley in 1858 stated the result for {{math|3&thinsp;×&thinsp;3}} and smaller matrices, but only published a proof for the {{math|2&thinsp;×&thinsp;2}} case.<ref>{{harvnb|Cayley|1858|pp=17–37}}</ref><ref>{{harvnb|Cayley|1889|pp=475–496}}</ref> As for {{math|''n''&thinsp;×&thinsp;''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”. The general case was first proved by [[Ferdinand Georg Frobenius|Ferdinand Frobenius]] in 1878.<ref name="Frobenius 1878">{{harvnb|Frobenius|1878}}</ref>