Page 39 - Matrices theory and applications

2. Square Matrices
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Corollary 2.3.1 Given an alternate matrix M ∈ M n (K),there exists a
matrix Q ∈ GL n (K) such that
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(2.2)
M = Q diag(J,... ,J, 0,... , 0)Q.
Obviously, the rank of M, being the same as that of the block-diagonal
matrix, equals twice the number of J blocks. Finally, since det J =1, we
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have det M = (det Q) ,where = 0 if there is a zero diagonal block in the
decomposition, and = 1 otherwise. Thus we have proved the following
result.
Proposition 2.3.2 The rank of an alternate matrix M is even. The num-
ber of J blocks in the identity (2.2) is the half of that rank. In particular,
it does not depend on the decomposition. Finally, the determinant of an
alternate matrix is a square in K.
A very important application of Proposition 2.3.2 concerns the Pfaffian,
whose crude definition is a polynomial whose square is the determinant of
the general alternate matrix. First of all, since the rank of an alternate
matrix is even, det M = 0 whenever n is odd. Therefore, we restrict our
attention from now on to the even-dimensional case n =2m. Let us consider
the field F = QQ(x ij ) of rational functions with rational coefficients, in
n(n − 1)/2 indeterminates x ij , i< j. We apply the proposition to the
alternate matrix X whose (i, i)-entry is 0 and (i, j)-entry (respectively (j, i)-
entry) is x ij (respectively −x ij ). Its determinant, a polynomial in ZZ[x ij ], is
the square of some irreducible rational function f/g,where f and g belong
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to ZZ[x ij ]. From g det X = f ,we see that g divides f in ZZ[x ij ]. But since
f and g are coprime, one finds that g is invertible; in other words g = ±1.
Thus
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det X = f . (2.3)
Now let k be a field and let M ∈ M n (k) be alternate. There exists
a unique homomorphism from ZZ[x ij ]into k sending x ij to m ij .From
equation (2.3) we obtain
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det M =(f(m 12 ,... ,m n−1,n )) . (2.4)
In particular, if k = QQ and M =diag(J,... ,J), one has f 2 =1. Up
to multiplication by ±1, which leaves unchanged the identity (2.3), we
may assume that f = 1 for this special case. This determination of the
polynomial f is called the Pfaffian and is denoted by Pf. It may be viewed
as a polynomial function on the vector space of alternate matrices with
entries in a given field k. equation (2.4) now reads
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det M =(Pf(M)) . (2.5)
Given an alternate matrix M ∈ M n (k)and amatrix Q ∈ M n (k), we
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consider the Pfaffian of the alternate matrix Q MQ. We first consider the
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case of the field of fractions QQ(x ij ,y ij )inthe n +n(n−1)/2 indeterminates