Annihilating polynomial

An annihilating polynomial for a matrix is a polynomial whose value for a given square matrix is equal to the zero matrix. The Hamilton-Cayley theorem states that the value of the characteristic polynomial for a square matrix is equal to the zero matrix, which means that for each square matrix there is at least one annihilating polynomial of degree coinciding with the order of the matrix .

An annihilating polynomial for a vector is a polynomial whose value for a given square matrix and a given vector is equal to zero vector. In other words, the polynomial $f$ ${\ displaystyle f}$ $f$ is invalidating for the matrix $A$ ${\ displaystyle A}$ $A$ and vectors $x$ ${\ displaystyle x}$ $x$ , if a $f(A)(x)={\overline {0}}$ ${\ displaystyle f (A) (x) = {\ overline {0}}}$ $f (A) (x) = \ overline 0$ . According to the definition of the kernel , this is the same as $x\in \ker f(A)$ ${\ displaystyle x \ in \ ker f (A)}$ $x \ in \ ker f (A)$ .

Literature

Gantmakher F.R. Matrix Theory (2nd ed.). M .: Science , 1966
Lancaster P. Matrix Theory M .: Science , 1973
Prasolov V.V. Problems and theorems of linear algebra. - M .: Nauka, 1996 .-- 304 p. - ISBN 5-02-014727-3 .

Annihilating polynomial

Literature

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