By Aaron Marcus
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Additional resources for An Introduction to Homological Algebra
Then the subspaces Wi are invariant under T, and V is their direct sum : V = W1 ⊕ W2 ⊕ · · · ⊕ Wk . Proof In essence, the proof is the same one used to establish the primary decomposition of a ﬁnite abelian group into p-groups. 1, the Wi are T-invariant. i For each i, let fi = p/pm i . Since these polynomials are relatively prime, there exist polynomials g1 , g2 , . . , gk such that f1 g1 + f2 g2 + · · · + fk gk = 1. It follows that f1 (T)g1 (T) + f2 (T)g2 (T) + · · · + fk (T)gk (T) = I . Now given v ∈ V , we have v = I(v) = ki=1 fi (T)gi (T)(v) with fi (T)gi (T)(v) ∈ Wi because pi (T)mi fi (T)gi (T)(v) = p(T)gi (T)(v) = 0.
If A is nilpotent then 0 is its only eigenvalue. 1 Over an algebraically closed ﬁeld, an n × n matrix A is nilpotent if and only if 0 is the only eigenvalue of A. Also, a square matrix that does not have two distinct eigenvalues must be the sum of a scalar matrix λI and a nilpotent matrix. Proof The second statement follows from the ﬁrst because if λ is the only eigenvalue of A, then 0 is the only eigenvalue of A − λI (and A = λI + (A − λI) ). Suppose 0 is the only eigenvalue of A. Since the ﬁeld is algebraically closed, the characteristic polynomial of A must be p(x) = xn .
A good canonical form under similarity allows an indirect way of seeing if two given matrices A, B ∈ Mn (F) are similar, by checking if they have the same canonical form. This is sometimes highlighted by authors as the raison d’être for having a canonical form. That the present authors chose not to list this outcome among their ﬁve desirable features of a canonical form would suggest they disagree ! Nevertheless, the Jordan and Weyr forms both perform well in this method of testing for similarity of matrices A and B (with perhaps Weyr a little more transparent), providing one knows the eigenvalues of A and B (which must agree for similar matrices), and also knows the nullities of the various powers (λI − A)i and (λI − B)i for i = 1, 2, .
An Introduction to Homological Algebra by Aaron Marcus