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| {{for|the minimal polynomial of an algebraic element of a field|Minimal polynomial (field theory)}}
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| In [[linear algebra]], the '''minimal polynomial''' {{math|''μ''<sub>''A''</sub>}} of an ''n''-by-''n'' [[matrix (mathematics)|matrix]] ''A'' over a [[field (mathematics)|field]] '''F''' is the [[monic polynomial]] ''P'' over '''F''' of least degree such that ''P''(''A'')=0. Any other polynomial ''Q'' with ''Q''(''A'') = 0 is a (polynomial) multiple of {{math|''μ''<sub>''A''</sub>}}.
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| The following three statements are equivalent:
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| # ''λ'' is a root of {{math|''μ''<sub>''A''</sub>}},
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| # ''λ'' is a root of the [[characteristic polynomial]] {{math|''χ''<sub>''A''</sub>}} of ''A'',
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| # ''λ'' is an [[eigenvalue]] of matrix ''A''.
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| The multiplicity of a root ''λ'' of {{math|''μ''<sub>''A''</sub>}} is the largest power ''m'' such that {{math|Ker((''A'' − ''λI<sub>n</sub>'')<sup>''m''</sup>)}} ''strictly'' contains {{math|Ker((''A'' − ''λI<sub>n</sub>'')<sup>''m''−1</sup>)}} (increasing the exponent up to ''m'' will give ever larger kernels, but further increasing ''m'' will just give the same kernel).
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| If the field {{math|''F''}} is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in {{math|''F''}}) alone, in other words they may have [[irreducible polynomial]] factors of degree greater than 1. For irreducible polynomials {{math|''P''}} one has similar equivalences:
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| # {{math|''P''}} divides {{math|''μ''<sub>''A''</sub>}},
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| # {{math|''P''}} divides {{math|''χ''<sub>''A''</sub>}},
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| # the kernel of {{math|''P''(''A'')}} has dimension at least 1.
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| # the kernel of {{math|''P''(''A'')}} has dimension at least deg(''P'').
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| 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 than 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 ''A'': extending the base field will not introduce any new such relations (nor of course will it remove existing ones).
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| The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if ''A'' is a multiple <math>aI_n</math> of the identity matrix, then its minimal polynomial is {{math|''X'' − ''a''}} since the kernel of <math>aI_n-A=0</math> is already the entire space; on the other hand its characteristic polynomial is {{math|(''X'' − ''a'')<sup>''n''</sup>}} (the only eigenvalue is {{math|''a''}}, 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).
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| == Formal definition ==
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| Given an [[endomorphism]] ''T'' on a finite-dimensional [[vector space]] ''V'' over a [[Field (mathematics)|field]] '''F''', let ''I''<sub>''T''</sub> be the set defined as
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| :<math> \mathit{I}_T = \{ p \in \mathbf{F}[t] \; | \; p(T) = 0 \} </math>
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| where '''F'''[''t''] is the space of all polynomials over the field '''F'''. ''I''<sub>''T''</sub> is a [[Ideal (ring theory)|proper ideal]] of '''F'''[''t''].
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| *The '''minimal polynomial''' is the [[monic polynomial]] which generates ''I''<sub>''T''</sub>.
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| Thus it must be the monic polynomial of least degree in ''I''<sub>''T''</sub>.
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| == Applications ==
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| An [[endomorphism]] ''φ'' of a finite dimensional vector space over a field '''F''' is [[diagonalizable]] if and only if its minimal polynomial factors completely over '''F''' into ''distinct'' linear factors. The fact that there is only one factor {{math|''X'' − ''λ''}} for every eigenvalue {{mvar|''λ''}} means that the [[generalized eigenspace]] for {{mvar|''λ''}} is the same as the [[eigenspace]] for {{mvar|''λ''}}: every Jordan block has size 1. More generally, if ''φ'' satisfies a polynomial equation {{math|''P''(''φ'') {{=}} 0}} where ''P'' factors into distinct linear factors over '''F''', then it will be diagonalizable: its minimal polynomial is a divisor of ''P'' and therefore also factors into distinct linear factors. In particular one has:
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| * <math>P=X^k-1</math>: finite order endomorphisms of complex vector spaces are diagonalizable. For the special case {{math|''k'' = 2}} of [[involution (mathematics)|involutions]], this is even true for endomorphisms of vector spaces over any field of [[characteristic (algebra)|characteristic]] other than 2, since <math>X^2-1=(X-1)(X+1)</math> is a factorization into distinct factors over such a field. This is a part of [[representation theory]] of cyclic groups.
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| * <math>P=X^2-X=X(X-1)</math>: endomorphisms satisfying <math>\varphi^2=\varphi</math> are called [[Projection (linear algebra)|projections]], and are always diagonalizable (moreover their only eigenvalues are 0 and 1).
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| * By contrast if <math>\mu_\varphi=X^k</math> with {{math|''k'' ≥ 2}} then ''φ'' (a nilpotent endomorphism) is not necessarily diagonalizable, since <math>X^k</math> has a repeated root 0.
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| These case can also be proved directly, but the minimal polynomial gives a unified perspective and proof.
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| == Computation ==
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| Let ''I''<sub>''T'',''v''</sub> be defined as
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| :::<math> \mathit{I}_{T, v} = \{ p \in \mathbf{F}[t] \; | \; p(T)(v) = 0 \},</math>
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| for some vectors ''v'' in ''V''. This definition satisfies the properties of a proper ideal. Let μ<sub>''T'',''v''</sub> be the monic polynomial which generates it.
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| ===Properties===
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| {{unordered list
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| |1= Since ''I''<sub>''T,v''</sub> contains minimal polynomial {{math|''μ''<sub>''T''</sub>}}, the latter is divisible by {{math|''μ''<sub>''T'',''v''</sub>}}.
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| |2= If ''d'' is the least natural number such that ''v'', ''T''(''v''), ... , ''T''<sup>''d''</sup>(''v'') are [[linearly dependent]], then there exist unique <math>a_0, a_1, \cdots, a_{d-1} \in \mathbf{F}</math> such that
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| :<math> a_0 v + a_1 T(v) + \cdots + a_{d-1} T^{d-1} (v) + T^d (v) = 0</math> | |
| and for these coefficients one has
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| <math> \mu_{T,v} (t) = a_0 + a_1 t + \ldots + a_{d-1} t^{d-1} + t^d. </math>
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| |3= Let the subspace ''W'' be the image of {{math|''μ''<sub>''T'',''v''</sub>}}(''T''), which is ''T''-stable. Since {{math|''μ''<sub>''T'',''v''</sub>}}(''T'') annihilates at least the vectors ''v'', ''T''(''v''), ... , ''T''<sup>''d''-1</sup>(''v''), the [[codimension]] of ''W'' is at least ''d''.
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| |4= The minimal polynomial {{math|''μ''<sub>''T''</sub>}} is the product of {{math|''μ''<sub>''T'',''v''</sub>}} and the minimal polynomial ''Q'' of the restriction of ''T'' to ''W''. In the (likely) case that ''W'' has dimension 0 one has {{math|''Q'' {{=}} 1}} and therefore {{math|''μ''<sub>''T''</sub> {{=}} ''μ''<sub>''T'',''v''</sub>}}; otherwise a recursive computation of ''Q'' suffices to find {{math|''μ''<sub>''T''</sub>}}.
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| }}
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| === Example ===
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| Define {{math|''T''}} to be the endomorphism of {{math|'''R'''<sup>3</sup>}} with matrix, on the canonical basis,
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| :<math>\begin{bmatrix} 1 & -1 & -1 \\ 1 & -2 & 1 \\ 0 & 1 & -3 \end{bmatrix}.</math> | |
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| Taking the first canonical basis vector {{math|''e''<sub>1</sub>}} and its repeated images by {{math|''T''}} one obtains
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| :<math> e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad
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| T\cdot e_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}. \quad
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| T^2\cdot e_1 = \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \mbox{ and}\quad
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| T^3\cdot e_1=\begin{bmatrix} 0 \\ 3 \\ -4 \end{bmatrix}</math>
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| of which the first three are easily seen to be [[linearly independent]], and therefore span all of {{math|'''R'''<sup>3</sup>}}. The last one then necessarily is a linear combination of the first three, in fact <math>T^3\cdot e_1 = -4T^2\cdot e_1 -T\cdot e_1 + e_1</math>, so that <math>\mu_{T,e_1}=X^3+4X^2+X-1</math>. This is in fact also the minimal polynomial <math>\mu_T</math> and the characteristic polynomial <math>\chi_T</math>: indeed <math>\mu_{T,e_1}</math> divides <math>\mu_T</math> which divides <math>\chi_T</math>, and since the first and last are of degree 3 and all are monic, they must all be the same. Another reason is that in general if any polynomial in {{math|''T''}} annihilates a vector {{math|''v''}}, then it also annihilates {{math|''T''⋅''v''}} (just apply {{math|''T''}} to the equation that says that it annihilates {{math|''v''}}), and therefore by iteration it annihilates the entire space generated by the iterated images by {{math|''T''}} of {{math|''v''}}; in the current case we have seen that for {{math|''v'' {{=}} ''e''<sub>1</sub>}} that space is all of {{math|'''R'''<sup>3</sup>}}, so <math>\mu_{T,e_1}(T)=0</math>. Indeed one verifies for the full matrix that <math>T^3+4T^2+T-I_3</math> is the null matrix:
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| :<math>\begin{bmatrix} 0 & 1 & -3 \\ 3 & -13 & 23 \\ -4 & 19 & -36 \end{bmatrix}
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| +4\begin{bmatrix} 0 & 0 & 1 \\ -1 & 4 & -6 \\ 1 & -5 & 10 \end{bmatrix}
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| +\begin{bmatrix} 1 & -1 & -1 \\ 1 & -2 & 1 \\ 0 & 1 & -3 \end{bmatrix}
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| +\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}
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| =\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
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| </math>
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| ==References==
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| * {{Lang Algebra}}
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| [[Category:Matrix theory]]
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| [[Category:Polynomials]]
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Yoshiko is her title but she doesn't like when people use her complete title. Interviewing is how I make a residing and it's some thing I truly enjoy. He currently life in Idaho and his mothers and fathers live nearby. Playing croquet is some thing I will never give up.
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