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In linear algebra, the minimal polynomial of an matrix over a field is the monic polynomial over of least degree such that . Any other polynomial with is a (polynomial) multiple of . The following three statements are equivalent: # is a root of , # is a root of the characteristic polynomial of , # is an eigenvalue of matrix . The multiplicity of a root of is the largest power such that ''strictly'' contains . In other words, increasing the exponent up to will give ever larger kernels, but further increasing the exponent beyond will just give the same kernel. If the field is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in ) alone, in other words they may have irreducible polynomial factors of degree greater than . For irreducible polynomials one has similar equivalences: # divides , # divides , # the kernel of has dimension at least . # the kernel of has dimension at least . Like the characteristic polynomial, the minimal polynomial does not depend on the base field, in other words considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason is somewhat different from for the characteristic polynomial (where it is immediate from the definition of determinants), namely the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of : extending the base field will not introduce any new such relations (nor of course will it remove existing ones). The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if is a multiple of the identity matrix, then its minimal polynomial is since the kernel of is already the entire space; on the other hand its characteristic polynomial is (the only eigenvalue is , and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field). == Formal definition == Given an endomorphism on a finite-dimensional vector space over a field , let be the set defined as : where is the space of all polynomials over the field . is a proper ideal of . Since is a field, is a principal ideal domain, thus any ideal is generated by a single polynomial, which is unique up to units in . A particular choice among the generators can be made, since precisely one of the generators is monic. The minimal polynomial is thus defined to be the monic polynomial which generates . It is the monic polynomial of least degree in . 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「Minimal polynomial (linear algebra)」の詳細全文を読む スポンサード リンク
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