By A. I. Kostrikin, I. R. Shafarevich
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Extra resources for Algebra II - Noncommunicative Rings, Identities
Xk} is A invariant. 1, and for some transformations not all invariant subspaces are of this form. 1 only one of the n nonzero invariant subspaces is spanned by eigenvectors. 2 every nonzero vector is an eigenvector corresponding to A 0 , so obviously every ^-invariant subspace is spanned by eigenvectors. 3 JORDAN CHAINS We have seen in the description of two-dimensional invariant subspaces that eigenvectors alone are not always sufficient for description of all invariant subspaces. This fact necessitates consideration of generalized eigenvectors as well.
3 Let XQ, . . , xk be a Jordan chain of A\M corresponding to the eigenvalue A0 of A\M. Then JCQ, . . , xk is also a Jordan chain of A corresponding to A 0 . In particular, all eigenvalues of A\M are also eigenvalues of A. Proof. We have x0 ^ 0; jr.. £ M for i = 0,. . , A;, and 24 Invariant Subspaces these relations can be rewritten as As But PXj — x^ i = 0,1,. . , k, and we obtain the relations defining JCQ, . . , xk as a Jordan chain of A corresponding to A0. 3 can also be proved in the following way.
Proof. If jV = JVR, we have already checked that ^V is angular. To prove the converse, assume that Jf is angular with respect to TT, and let Q be the projector of
Ker TT, and we have to show that N = NR. If x G jVw, then for some y = Try, Angular Transformations and Matrix Quadratic Equations Thus NR C N. Conversely, if y 6 N then thus N = NR, as required. 2). 1). Let _ y £ l m 7 r and x = Ry + y E. N. Then, since / - Q is onto Ker TT along jV But QR = Q and QTT = 0 so that Ry = (Q - ir)y.
Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich