Basics of the minimal polynomial最小多項式さいしょうたこうしき

date2026-07-14document_iddoc_e78fe37c2c8111c279b1dc10ff00a436description最小多項式を、行列を零化する多項式のうち最も基本的なものとして導入し、対角化可能性やジョルダン標準形との関係を整理する講義である。prerequisites固有値と固有ベクトル / 対角化の基本 / 行列式type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

The key point of this lecture is that the minimal polynomial最小多項式さいしょうたこうしき records the smallest algebraic relation satisfied by a matrix行列ぎょうれつ.

eigenvalues固有値こゆうち summarize important information, but they do not fully determine diagonalizability. The minimal polynomial最小多項式さいしょうたこうしき detects whether repeated factors are needed to annihilate the matrix, and that is exactly what distinguishes some matrices with the same eigenvalues固有値こゆうち.

2Terms and definitions

The minimal polynomial最小多項式さいしょうたこうしき of a square matrix正方行列せいほうぎょうれつ $A$ is the nonzero monic polynomial of least degree such that

m_{A} (A) = 0.

Let the characteristic polynomial特性多項式とくせいたこうしき be

χ_{A} (t) = \det (t I - A) .

By the Cayley-Hamilton theorem,

χ_{A} (A) = 0.

Therefore the minimal polynomial最小多項式さいしょうたこうしき exists and divides the characteristic polynomial特性多項式とくせいたこうしき.

3Plan

First understand what it means to substitute a matrix into a polynomial. Then relate the roots of the minimal polynomial最小多項式さいしょうたこうしき to the eigenvalues固有値こゆうち of $A$ , and finally use repeated roots to test diagonalizability.

data/lecture/math/linear-algebra/eigenvalues-and-eigenvectors.lecture.n.md

4Intuitive explanation

For a number $λ$ , the factor $t - λ$ vanishes when $t = λ$ . For a matrix $A$ , the expression $A - λ I$ usually does not become the zero matrix. But a suitable polynomial in $A$ can become zero.

The minimal polynomial最小多項式さいしょうたこうしき is the shortest such relation. If $A$ is diagonalizable, one factor for each distinct eigenvalue固有値こゆうち is enough. If $A$ is not diagonalizable, the same factor must sometimes appear more than once.

5Precise explanation

5.11. Relation with the characteristic polynomial特性多項式とくせいたこうしき

The minimal polynomial最小多項式さいしょうたこうしき divides the characteristic polynomial特性多項式とくせいたこうしき:

m_{A} (t) ∣ χ_{A} (t) .

Its roots are eigenvalues固有値こゆうち of $A$ , and every eigenvalue固有値こゆうち appears as a root of $m_{A} (t)$ .

5.22. Diagonalizability

Over a field $K$ , if the minimal polynomial最小多項式さいしょうたこうしき factors as

m_{A} (t) = \prod_{i = 1}^{r} (t - λ_{i})

with distinct linear factors, then $A$ is diagonalizable. Conversely, if $A$ is diagonalizable, the minimal polynomial最小多項式さいしょうたこうしき has this form.

Thus

[PARSE ERROR: Undefined("Command(\"boxed\")")] A isdiagonalizable ⟺ m_{A} (t) hasnorepeatedrootandsplitsintolinearfactors .

6Existence and uniqueness

Existence follows from Cayley-Hamilton: $χ_{A} (A) = 0$ , so at least one nonzero annihilating polynomial exists.

Uniqueness follows from choosing the monic annihilating polynomial of smallest degree. If two monic polynomials $m_{1}, m_{2}$ of that least degree both annihilate $A$ , then $m_{1} - m_{2}$ also annihilates $A$ and has smaller degree. This contradicts minimality unless $m_{1} = m_{2}$ .

7Proof idea for the squarefree criterion

If $A = P D P^{- 1}$ is diagonalizable, then for any polynomial $f$ ,

f (A) = P f (D) P^{- 1} .

The matrix $f (D)$ is diagonal with entries $f (λ_{i})$ . Therefore it is enough for $f$ to vanish on the distinct eigenvalues固有値こゆうち. Hence

m_{A} (t) = \prod_{λ \in Spec (A)} (t - λ),

with no repeated roots.

Here $Spec (A)$ denotes the set of eigenvalues固有値こゆうち of $A$ .

Conversely, if $m_{A} (t)$ splits into distinct linear factors, one can construct polynomial projections onto the eigenspaces. The space becomes a direct sum of eigenspaces, so an eigenvector固有こゆうベクトル basis基底きてい exists and $A$ is diagonalizable.

These projections are not necessarily orthogonal projections coming from an inner product. They are linear maps that extract the component lying in each eigenspace.

8Concrete examples

8.1Diagonal matrix

For

A = (\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}),

we have

(A - 2 I) (A - 3 I) = 0,

so

m_{A} (t) = (t - 2) (t - 3) .

8.2Jordan blockジョルダンブロック

As preparation for the next lecture, this smallest non-diagonalizable example is called a Jordan block here.

For

B = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}),

the matrix

B - I = (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})

is not zero, but

(B - I)^{2} = 0.

Therefore

m_{B} (t) = (t - 1)^{2},

and $B$ is not diagonalizable.

8.3Same characteristic polynomial特性多項式とくせいたこうしき, different minimal polynomials

The matrices

A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), B = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})

both have characteristic polynomial特性多項式とくせいたこうしき $(t - 1)^{2}$ . But

m_{A} (t) = t - 1, m_{B} (t) = (t - 1)^{2.}

The minimal polynomial最小多項式さいしょうたこうしき detects the non-diagonalizable Jordan behavior that the eigenvalue固有値こゆうち list alone does not.

9Another viewpoint: reducing powers and Jordan form

The minimal polynomial最小多項式さいしょうたこうしき is an algebraic relation for the algebra generated by the linear transformation線型変換せんけいへんかん. For example, once $m_{A} (t)$ is known, high powers such as $A^{k}$ can be reduced to lower powers.

When Jordan form is available, the exponent of $(t - λ)$ in $m_{A} (t)$ equals the size of the largest Jordan blockジョルダンブロック for $λ$ . The characteristic polynomial特性多項式とくせいたこうしき counts the total sizes of all blocks, while the minimal polynomial最小多項式さいしょうたこうしき records the maximum block length for each eigenvalue固有値こゆうち.

data/lecture/math/linear-algebra/introduction-to-jordan-canonical-form.lecture.n.md

The companion matrix同伴行列どうはんぎょうれつ constructs a matrix with a prescribed characteristic polynomial and connects this lecture to polynomial-driven matrix examples.

data/lecture/math/linear-algebra/companion-matrix-basics.lecture.n.md

10Another viewpoint

The minimal polynomial最小多項式さいしょうたこうしき is the defining algebraic relation for the algebra generated by a linear transformation. Knowing $m_{A} (t)$ allows high powers of $A$ to be reduced to lower powers.

11Criteria

$m_{A} (t)$ is the monic polynomial of least degree satisfying $m_{A} (A) = 0$ .
$m_{A} (t)$ divides the characteristic polynomial特性多項式とくせいたこうしき.
Diagonalizability corresponds to a squarefree minimal polynomial最小多項式さいしょうたこうしき that splits.
In Jordan form, the largest block size determines the exponent in the minimal polynomial最小多項式さいしょうたこうしき.

12Scope and limitations

Statements involving linear factors require the polynomial to split over the field being used. A polynomial that does not split over the real numbers may split after extending scalars to the complex numbers.

13Existence and uniqueness

The minimal polynomial最小多項式さいしょうたこうしき always exists and is unique. Existence follows from the Cayley-Hamilton theorem, because the characteristic polynomial特性多項式とくせいたこうしき $χ_{A} (t)$ satisfies

χ_{A} (A) = 0.

Thus at least one nonzero polynomial annihilating $A$ exists.

Uniqueness is obtained by choosing the monic annihilating polynomial of least degree. If two monic polynomials $m_{1}, m_{2}$ of the same least degree both annihilate $A$ , then $m_{1} - m_{2}$ is an annihilating polynomial of smaller degree. This contradicts minimality unless $m_{1} = m_{2}$ .

14Proof idea: no repeated roots and diagonalization対角化たいかくか

Suppose $A$ is diagonalizable and $A = P D P^{- 1}$ , where $D = diag (λ_{1}, [PARSE ERROR: Undefined("Command(\"dots\")")], λ_{n})$ . For any polynomial $f$ ,

f (A) = P f (D) P^{- 1},

and $f (D)$ is the diagonal matrix with entries $f (λ_{i})$ . Therefore, to annihilate $A$ , it is enough for $f$ to vanish at every eigenvalue固有値こゆうち that appears. Hence

m_{A} (t) = \prod_{λ \in Spec (A)} (t - λ),

so $m_{A} (t)$ has no repeated root.

Conversely, if $m_{A} (t)$ splits as a product of distinct linear factors, polynomial projections onto the eigenspaces can be constructed. The space decomposes as a direct sum of eigenspaces固有空間こゆうくうかん, so an eigenvector固有こゆうベクトル basis基底きてい exists and $A$ is diagonalizable.

15Correspondence with Jordan normal formジョルダン標準形ひょうじゅんけい

When Jordan normal form is available, the minimal polynomial最小多項式さいしょうたこうしき records, for each eigenvalue固有値こゆうち, the size of the largest Jordan block. For example, if the largest block for $λ$ has size $3$ , then $(t - λ)^{3}$ appears in $m_{A} (t)$ .

Thus the characteristic polynomial特性多項式とくせいたこうしき counts the total amount of blocks, while the minimal polynomial最小多項式さいしょうたこうしき measures the maximum block length. Diagonalizability is exactly the case where every maximum block length is $1$ .