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ÕªÒª£º Hamilton-Cayley ¶¨Àí Éè \(A\in F^{n\times n}\), \(C(\lambda)=|\lambda I-A|\)£¬Ôò$C(A)=O$. Éè \(f\in Hom(V)\)£¬\(C(\lambda)\) ÊÇ \(f\) µÄÌØÕ÷¶àÏîʽ£¬Ôò \(C(f)=O\). ÌâÄ¿ ÔĶÁÈ«ÎÄ
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2020Äê5ÔÂ19ÈÕ

ÕªÒª£º ÃüÌâ Éè $f(x)$ ÊǶàÏîʽ¡£Èô $f(A)=O$£¬Ôò $A$ µÄÌØÕ÷Öµ¾ùÊÇ $f(x)=0$ µÄ¸ù¡£ Ö¤Ã÷ ¶Ô $A$ µÄÌØÕ÷Öµ $\lambda_0$ ºÍÌØÕ÷ÏòÁ¿ $\eta, \eta\ne \theta$£¬ÓÐ $$ A\eta = \lambda_0\eta $$ $$ A^2\eta ÔĶÁÈ«ÎÄ
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ÕªÒª£º ¶¨Àí Éè $A=(a_{ij})_{n\times n}$£¬Ôò $$ |\lambda I A|= \lambda^n + b_1\lambda^{n 1} +\cdots+b_{n 1}\lambda + b_n $$ ÆäÖУ¬ $b_j=( 1)^j\sum (AµÄj½×Ö÷×Óʽ)$. ÌرðµØ£¬$b_ ÔĶÁÈ«ÎÄ
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2020Äê5ÔÂ17ÈÕ

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