Page 406 - Matrix Analysis & Applied Linear Algebra

402 Chapter 5 Norms, Inner Products, and Orthogonality

5.10.11. An algebraic group is a set G together with an associative operation
between its elements such that G is closed with respect to this operation;
G possesses an identity element E (which can be proven to be unique);
and every member A ∈G has an inverse A # (which can be proven to
be unique). These are essentially the axioms (A1), (A2), (A4), and (A5)
in the definition of a vector space given on p. 160. A matrix group is
a set of square matrices that forms an algebraic group under ordinary
matrix multiplication.
(a) Show that the set of n × n nonsingular matrices is a matrix
group.
(b) Show that the set of n × n unitary matrices is a subgroup of
the n × n nonsingular matrices.
5 α α 6
(c) Show that the set G = α =0 is a matrix group.

α α
In particular, what does the identity element E ∈G look like,
and what does the inverse A # of A ∈G look like?
5.10.12. For singular matrices, prove that the following statements are equivalent.
(a) A is a group matrix (i.e., A belongs to a matrix group).
(b) R (A) ∩ N (A)= 0.
(c) R (A) and N (A) are complementary subspaces.
(d) index(A)=1.
(e) There are nonsingular matrices Q n×n and C r×r such that

0
−1
C r×r
Q AQ = , where r = rank (A).
0 0
5.10.13. Let A ∈G for some matrix group G.
(a) Show that the identity element E ∈G is the projector onto
R (A) along N (A)by arguing that E must be of the form
0
E = Q I r×r Q −1 .
0 0
(b) Show that the group inverse of A (the inverse of A in G )
must be of the form
−1
C 0 −1
#
A = Q Q .
0 0