关于凯莱-哈密顿定理的澄清及相关多项式性质问询

Great question! Let's unpack all the key conclusions we can draw when a polynomial $p(T) \in R[T]$ annihilates a square matrix $A$ over a commutative ring $R$, beyond the basics you already noted.

1. Core Divisibility by the Minimal Polynomial #

• As you pointed out, the minimal polynomial $m_A(T)$ (the monic polynomial of smallest degree such that $m_A(A) = 0$) divides $p(T)$ in $R[T]$. This is the foundational algebraic relationship: every polynomial that annihilates $A$ must be a multiple of the minimal polynomial.

2. Shared Irreducible Factors with the Characteristic Polynomial #

• By Cayley-Hamilton, the characteristic polynomial $\chi_A(T)$ also annihilates $A$, so $m_A(T) \mid \chi_A(T)$ too. This means every irreducible factor of $\chi_A(T)$ (over $R$) must be a factor of $p(T)$. In other words, $p(T)$ can't miss any irreducible component of the characteristic polynomial—it has to "cover" all the algebraic structure encoded in $\chi_A(T)$.

3. Primary Decomposition Constraints (For PIDs like $\mathbb{Z}$ or fields) #

• If $R$ is a Principal Ideal Domain (PID), we can use the structure theorem for finitely generated modules over PIDs. When we view $R^n$ as an $R[T]$-module where $T$ acts via $A$, the annihilator of this module is exactly the ideal generated by $m_A(T)$.
  • For the primary decomposition of $m_A(T)$—say $m_A(T) = \prod_{i} f_i(T)^{k_i}$ where each $f_i$ is irreducible—$p(T)$ must be divisible by each $f_i(T)^{k_i}$. This means for every irreducible factor $f$ of $m_A$, the exponent of $f$ in $p(T)$ is at least as large as its exponent in $m_A(T)$.

4. Jordan Block Size Requirements (For Algebraically Closed Fields) #

• If $R$ is an algebraically closed field (like $\mathbb{C}$), $A$ has a Jordan canonical form. For each Jordan block of size $s$ corresponding to eigenvalue $\lambda$, the minimal polynomial includes $(T - \lambda)^s$ as a factor. So $p(T)$ must also have $(T - \lambda)^s$ as a factor—this is stronger than just having $\lambda$ as a root: it accounts for the nilpotent "tail" of the Jordan block. In short, the multiplicity of $\lambda$ in $p(T)$ must be at least the size of the largest Jordan block for $\lambda$.

5. Ring-Theoretic Interpretation via Quotient Rings #

• The subring of $M_n(R)$ generated by $A$ is isomorphic to the quotient ring $R[T]/(m_A(T))$ (via the homomorphism sending $T$ to $A$). Since $p(A) = 0$, the image of $p(T)$ in this quotient ring is zero. This just rephrases the divisibility condition in ring terms, but it helps connect matrix annihilation to abstract algebra.

6. Coefficient Constraints for Specific Rings #

• For concrete rings like $\mathbb{Z}$ (integers), annihilating polynomials have to satisfy strict coefficient conditions. For example, if $A$ is an integer matrix with finite order $k$ (meaning $A^k = I$), then $p(T)$ must be divisible by $T^k - 1$, so $p(T) = (T^k - 1)q(T)$ for some integer polynomial $q(T)$. More generally, substituting $A$ into $p(T)$ must yield the zero matrix, which translates to integer congruence conditions on $p$'s coefficients relative to $A$'s entries.

内容的提问来源于stack exchange，提问作者Drew Brady