By James E. Humphreys
The e-book offers an invaluable exposition of effects at the constitution of semisimple algebraic teams over an arbitrary algebraically closed box. After the basic paintings of Borel and Chevalley within the Fifties and Sixties, additional effects have been got over the subsequent thirty years on conjugacy periods and centralizers of parts of such teams
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Extra resources for Conjugacy classes in semisimple algebraic groups
A2 .. Ar In this case, we write A = diag(A1 , A2 , . . , 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.
In particular, idempotent matrices have only 0 and 1 as eigenvalues (combined with diagonalizability, this characterizes idempotents). Idempotent linear transformations T : V → V are exactly the projection maps T : U ⊕ W −→ U , u + w −→ u associated with direct sum decompositions V = U ⊕ W . Necessarily, U is the image of T, on which T acts as the identity transformation, and W is its kernel, on which T acts, of course, as the zero transformation. 2 gives the displayed idempotent matrix E of T for a suitable basis.
The authors do not regard themselves as specialists in applied linear algebra. 12 ADVANCED TOPICS IN LINEAR ALGEBRA To keep the discussion simple, we will work with square matrices A over an arbitrary ﬁeld. We can partition the matrix A by choosing some horizontal partitioning of the rows and, independently, some vertical partitioning of the columns. For instance, ⎡ ⎢ ⎢ ⎢ ⎢ A = ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ combines the horizontal partitioning 6 = 3 + 1 + 1 + 1 with the vertical partitioning 6 = 1 + 1 + 3 + 1.