By Charles A. Weibel
A portrait of the topic of homological algebra because it exists this present day
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Additional info for An Introduction to Homological Algebra
Ar ) and say A is a direct sum of the matrices A1 , A2 , . . , Ar . If B = diag(B1 , B2 , . . , Br ) is a second block diagonal matrix (for the same blocking), then AB = diag(A1 B1 , A2 B2 , . . , Ar Br ). Of course, sums and scalar multiples behave similarly, so our knowledge of a block diagonal matrix is as good as our knowledge of its individual diagonal blocks. This is a simple but fundamental observation, used again and again in canonical forms, for instance. Those with a ring theory background may prefer to view this as saying the following.
However, under multiplication, the invertible n × n matrices form the general linear group GLn (F). Its rich group structure, harnessed through group representation theory and character theory, as developed by Frobenius, Burnside, Schur, Brauer and others, provides the most powerful tool known for describing ﬁnite groups and other classes of groups. There is also a sort of complementary multiplicative relationship of invertible and idempotent matrices. A 1967 result of J. A. Erdos says that every noninvertible matrix is a product of idempotent matrices.
Sylvester discovered the result in 1884. The theorem is well-known but deserves to be even better known, if for no other reason than its proof highlights the power of switching back and forth between matrices and linear transformations. 20. The operator version was ﬁrst noted in the late 1940’s and independently published by Dalecki in 1953 and Rosenblum in 1956. 34 ADVANCED TOPICS IN LINEAR ALGEBRA Let V = Mn×m (F) and regard V as an mn-dimensional vector space over F. Let TA : V → V and TB : V → V be the left and right multiplication maps by A and B, respectively: TA (X) = AX , TB (X) = XB for all X ∈ V .
An Introduction to Homological Algebra by Charles A. Weibel