Page 28 - Matrices theory and applications

1.2. Change of Basis
T
M = 0 amounts to writing x My =0 for every x and y.If m = n and
T
x Mx =0 for every x,one says that M is alternate. An alternate matrix
is skew-symmetric, since
T
T
T
T
T
x (M +M )y = x My+y Mx =(x+y) M(x+y)−x Mx−y My =0.
The converse holds whenever the characteristic of K is not 2, since
T
T
T
2x Mx = x (M + M )x =0. T T 11
However, in characteristic 2 there exist matrices that are skew-symmetric
but not alternate. As a matter of fact, the diagonal of an alternate matrix
must vanish, though this need not be the case for a skew-symmetric matrix
in characteristic 2.
The interpretation of transposition in terms of linear maps is the
n
following. One provides K with the bilinear form
T
T
x, y
:= x y = y x = x 1 y 1 + ··· + x n y n ,
m
called the canonical scalar product; one proceeds similarly in K .If M ∈
M n×m(K), there exists a unique matrix N ∈ M m×n (K) satisfying
Mx, y
= x, Ny
,
n
for all x ∈ K m and y ∈ K (notice that the scalar products are defined on
T
distinct vector spaces). One checks easily that N = M . More generally, if
E, F are K-vector spaces endowed with nondegenerate symmetric bilinear
T
forms, and if u ∈L(E, F), then one can define a unique u ∈L(F, E)from
the identity
T
u(x),y
F = x, u (y)
E , ∀x ∈ E, y ∈ F.
n
When E = K m and F = K are endowed with their canonical bases and
T
canonical scalar products, the matrix associated to u is the transpose of
the matrix associated to u.
Let K be a field. Let us endow K m with its canonical scalar product. If
m
F is a linear subspace of K , one defines the orthogonal subspace of F by
m
F ⊥ := {x ∈ K ; x, y
=0, ∀y ∈ F}.
m
It is a linear subspace of K . We observe that for a general field, the
intersection F ∩ F ⊥ can be nontrivial, and K m may differ from F + F .
⊥
One has nevertheless
dim F +dim F ⊥ = m.
Actually, F ⊥ is the kernel of the linear map T : K m →L(F; K)=: F ,
∗
m
defined by T (x)(y)= x, y
for x ∈ K , y ∈ F.Let us show that T is onto.
If {f 1,... ,f r } is a basis of F, then every linear form l on F is a map

f = z j f j → l(f)= l(f j )z j .
j j