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5.7. BILINEAR AND QUADRATIC FORMS 221

In terms of coordinates, (5.7.5.2) takes the form
◦

x k = x + x k (k = 1, 2, ... , n),
k
◦ T ◦ ◦
T

where X =(x 1 , ... , x n ), X T =(x , ... , x ), X =(x , ... , x ).
1
1
n
n
Under parallel translations any basis remains unchanged.
The transformation of the space V defined by (5.7.5.2) reduces the hypersurface equation
(5.7.5.1) to
A(x , x )+ 2B (x )+ c = 0,

where the linear form B (x ) and the constant c are defined by

◦ ◦ ◦ ◦

B (x )= A(x , x)+ B(x ), c = A(x, x)+ 2B(x)+ c,
or, in coordinate notation,
n n n
◦ ◦
B (x ) ≡ B X = b x , c = (b + b i )x + c, b = a ij j + b i .

x

i
i i
i
i
i=1 i=1 j=1
Under parallel translation the group of the leading terms preserves its form.
5.7.5-3. Transformation of one orthonormal basis into another.

The transition from one orthonormal basis i 1 , ... , i n to another orthonormal basis i , ... , i n
1
is defined by an orthogonal matrix P ≡ [p ij ]of size n × n, i.e.,
n

i = p ij i j (i = 1, 2, ... , n).
i
j=1
Under this orthogonal transformation, the coordinates of points are transformed as
follows:

X = PX,
or, in coordinate notation,
n

x = p ki x i (k = 1, 2, ... , n), (5.7.5.3)
k
i=1
T

where X =(x 1 , ... , x n ), X T =(x , ... , x ).
1
n
If the transition from the orthonormal basis i 1 , ... , i n to the orthonormal basis i , ... , i n

1
is defined by an orthogonal matrix P, then the hypersurface equation (5.7.5.1) in the new
basis takes the form

A (x , x )+ 2B (x )+ c = 0.

T
The matrix A ≡ [a ](A (x , x )= X A X ) is found from the relation

ij
–1

A = P AP.
Thus, when passing from one orthonormal basis to another orthonormal basis, the matrix
of a quadratic form is transformed similarly to the matrix of some linear operator. Note
that the operator A whose matrix in an orthonormal basis coincides with the matrix of the
quadratic form A(x, x) is Hermitian.