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Concept
Minimal polynomial (linear algebra)
Summary
In linear algebra, the minimal polynomial μA of an n × n matrix A over a 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 μA.
The following three statements are equivalent:
λ is a root of μA,
λ is a root of the characteristic polynomial χA of A,
λ is an eigenvalue of matrix A.
The multiplicity of a root λ of μA is the largest power m such that ker((A − λIn)m) strictly contains ker((A − λIn)m−1). In other words, increasing the exponent up to m will give ever larger kernels, but further increasing the exponent beyond m will just give the same kernel. Formally, m is the nilpotent index of A − λIn.
If the field F is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree greater than 1. For irreducible polynomials P one has similar equivalences:
P divides μA,
P divides χA,
the kernel of P(A) has dimension at least 1.
the kernel of P(A) has dimension at least deg(P).
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 for this differs from the case with the characteristic polynomial (where it is immediate from the definition of determinants), namely by 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).
The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if A is a multiple aIn of the identity matrix, then its minimal polynomial is X − a since the kernel of aIn − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a)n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the dimension of the space).
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