Last Updated：2025/11/28

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(linear algebra) For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.

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minimal polynomial

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Source Word

minimal polynomial

Noun

Japanese Meaning

与えられた正方行列に対して、その行列を0行列にする作用を持つ、係数が体K上にある最小次数の単調多項式（先頭係数が1）

What is this buttons?

To determine the minimal polynomial of a square matrix M over a field K, we look for the monic polynomial of least degree with coefficients in K that evaluates to the zero matrix when applied to M.

Google Translation / DeepL Translation

体K上の正方行列Mに対して、Mに代入すると零行列になる次数が最も小さい最高次係数が1の多項式を求めることで最小多項式を決定する。

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Sense（1）

Each root of the minimal polynomial of a matrix M is an eigenvalue of M and a root of its characteristic polynomial. (A root of the minimal polynomial has a multiplicity that is less than or equal to the multiplicity of the same root in the characteristic polynomial. Thus the minimal polynomial divides the characteristic polynomial. Also, any root of the characteristic polynomial is also a root of the minimal polynomial, so the two kinds of polynomial have the same roots, only (possibly) differing in their multiplicities.)

Google Translation / DeepL Translation