Page 50 - Matrices theory and applications

11. Compute the number of elements in the group GL 2 (ZZ/2ZZ). Show
that it is not commutative. Show that it is isomorphic to the
symmetric group S m , for a suitable integer m.
n
is given, one defines the circulant matrix
12. If (a 0 ,... ,a n−1 ) ∈ CC
circ(a 0 ,... ,a n−1) ∈ M n (CC)by
 2.9. Exercises 33

a 0 a 1 ... a n−1
. .
 . . 
a n−1 a 0
 . . 
circ(a 0 ,... ,a n−1):=   .
. . .
 . . . 
 . . . a 1 
a 1 ... a n−1 a 0
We denote by C n the set of circulant matrices. Obviously, the ma-
trix circ(1, 0, 0,... , 0) is the identity. The matrix circ(0, 1, 0,... , 0)
is denoted by π.
(a) Show that C n is a subalgebra of M n (CC), equal to CC[π]. Deduce
n
that it is isomorphic to the quotient ring CC[X]/(X − 1).
(b) Let C be a circulant matrix. Show that C ,aswell as P(C), is
∗
circulant for every polynomial P.If C is nonsingular, show that
C −1 is circulant.
(c) Show that the elements of C n are diagonalizable in a common
eigenbasis.
(d) Replace CC by any field K.If K contains a primitive nth root ω
n
of unity (that is, ω =1, and ω m = 1 implies m ∈ nZZ), show
that the elements of C n are diagonalizable.
Note: A thorough presentation of circulant matrices and
applications is given in Davis’s book [12].
(e) One assumes that the characteristic of K divides n. Show that
C n contains matrices that are not diagonalizable.
13. Show that the Pfaffian is linear with respect to any row or column
of an alternate matrix. Deduce that the Pfaffian is an irreducible
polynomial in ZZ[x ij ].
14. (Schur’s Lemma).
Let k be an algebraically closed field and S a subset of M n (k). As-
n
sume that the only linear subspaces of k that are stable under every
n
element of S are {0} and k itself. Let A ∈ M n (k) be a matrix that
commutes with every element of S. Show that there exists c ∈ k such
that A = cI n .
15. (a) Show that A ∈ M n (K) is irreducible if and only if for every pair
(j, k)with1 ≤ j, k ≤ n, there exists a finite sequence of indices
=0.
j = l 1 ,... ,l r = k such that a l p ,l p+1
(b) Show that a tridiagonal matrix A ∈ M n(K), for which none of
the a j,j+1 ’s and a j+1,j ’s vanish, is irreducible.