Page 474 - Advanced Linear Algebra
P. 474
458 Advanced Linear Algebra
Since the endomorphism algebras B - ²= ³ are of obvious importance, let us
examine them a bit more closely.
Theorem 18.3 Let be a vector space over a field .
=
-
)
1 The algebra B - ²= ³ has center
A~ ¸ -¹
and so B - ²= ³ is central.
) 0 ² = ³ that have finite rank is an ideal of
2 The set of all elements of B -
B B - - ²= ³ and is contained in all other ideals of ²= ³.
)
=
3 B - ²= ³ is simple if and only if is finite-dimensional.
Proof. We leave the proof of parts 1) and 3) as exercises. For part 2), we leave
0
1
it to the reader to show that is an ideal of B - ² = ³ . Let be a nonzero ideal of
B B - - ²= ³. Let ²= ³ have rank . Then there is a basis 8 ~ 8 r 8 (a
disjoint union) and a nonzero $ = for which 8 is a finite set, ² 8 ³ ~ ¸ ¹
and ² ³ ~ $ for all 8 . Thus, is a linear combination over - of
endomorphisms defined by
8
² ³ ~ $Á ² ± ¸ ¹³ ~ ¸ ¹
Hence, we need only show that 1 .
If 1 is nonzero, then there is an 8 for which ~ " £ . If B - ²= ³
is defined by
² ± ¸ ¹³ ~ ¸ ¹
8~ Á
and B ² = - ³ is defined by
8
" ~ $Á ² ± ¸"¹³ ~ ¸ ¹
then
²
² ³ ~ $Á 8 ± ¸ ¹³ ~ ¸ ¹
and so ~ . 1
Annihilators and Minimal Polynomials
If is an -algebra an ( , then it may happen that satisfies a nonzero
-
(
polynomial ²%³ -´%µ . This always happens, in particular, if ( is finite-
dimensional, since in this case the powers
Á Á Á Ã
must be linearly dependent and so there is a nonzero polynomial in that is
equal to .
Definition Let be an -algebra. An element ( is algebraic if there is a
-
(
nonzero polynomial ²%³ -´%µ for which ² ³ ~ . If is algebraic, the
