Page 40 - Matrices theory and applications

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2.4. Eigenvalues and Eigenvectors
x ij (1 ≤ i<j ≤ n)and y ij (1 ≤ i, j ≤ n). Let Y be the matrix whose
(i, j)-entry is y ij . Then, with X as above,
T
T
2
2
2
(Pf(Y XY )) =det Y XY =(det Y ) det X =(Pf(X)det Y ) .
Since ZZ[x ij ,y ij ] is an integral domain, we have the polynomial identity
T

Pf Y XY = Pf(X)det Y,
= ±1.
T
As above, one infers that Pf(Q MQ)= ± Pf(M)det Q for every field k,
matrix Q ∈ M n(k), and alternate matrix M ∈ M n (k). Inspection of the
particular case Q = I n yields = 1. We summarize these results now.
Theorem 2.3.1 Let n =2m be an even integer. There exists a unique
polynomial Pf in the indeterminates x ij (1 ≤ i<j ≤ n) with integer
coefficients such that:
• For every field k and every alternate matrix M ∈ M n (k), one has
2
det M =Pf(M) .
• If M = diag(J,... ,J),then Pf(M)= 1.
T
Moreover, if Q ∈ M n (k) is given, then Pf Q MQ =Pf(M)det Q.
We warn the reader that if m> 1, there does not exist a matrix Z ∈ QQ[x ij ]
T
such that X = Z diag(J,... ,J)Z. The factorization of the polynomial
det X does not correspond to a similar factorization of X itself. In other
T
words, the decomposition X = Q diag(J,... ,J)Q in M n (QQ(x ij )) cannot
be written within M n (QQ[x ij ]).
The Pfaffian is computed easily for small values of n. For instance,
Pf(X)= x 12 if n =2, and Pf = x 12 x 34 − x 13 x 24 + x 14 x 23 if n =4.
2.4 Eigenvalues and Eigenvectors
Let K be a field and E, F two vector spaces of finite dimension. Let us
recall that if u : E → F is a linear map, then
dim E =dim ker u +rk u,
where rk u denotes the dimension of u(E) (the rank of u). In particular, if
u ∈ End(E), then
u is bijective ⇐⇒ u is injective ⇐⇒ u is surjective.
However, u is bijective, that is invertible, in End(E), if and only if its
matrix M in some basis β is invertible, that is if its determinant is nonzero.
As a matter of fact, the matrix of u −1 is M −1 ; the existence of an inverse
(either that of M or that of u) implies that of the other one. Finally, if
M ∈ M n (K), then det M = 0 is equivalent to
n
∀X ∈ K , MX =0 =⇒ X =0.