8.6. RATIONAL CANONICAL FORM 403 that ± A .x/ is a similarity invariant for A ; that is, it is unchanged if A is re-placed by a similar matrix. Let V be an n –dimensional vector space over K and let T 2 End K .V / . If A is the matrix of T with respect to some basis of V , deﬁne ± T .x/ D ± A .x/ . It follows from the invariance of ± A under similarity that ± T is well–deﬁned (does not depend on the choice of basis) and that ± T is a similarity invariant for linear transformations. See Exercise 8.6.5 . ± T .x/ is called the characteristic polynomial of T . Let A be the matrix of T with respect to some basis of V . Consider the diagonalization of xE n ± A in Mat n .KŒxŁ/ , P.xE n ± A/Q D D.x/ D diag .1;1;:::;1;a 1 .x/;a 2 .x/;:::;a s .x//; where the a i .x/ are the (monic) invariant factors of T . We have ± T .x/ D ± A .x/ D det .xE n ± A/ D det .P ± 1 / det .D.x// det .Q ± 1 /: P ± 1 and Q ± 1 are invertible matrices in Mat n .KŒxŁ/ , so their determinants are units in KŒxŁ , that is nonzero elements of K
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