Page 246 -

5.7. BILINEAR AND QUADRATIC FORMS 213

5.6.3-6. Canonical form of linear operators.

Anelement xiscalledanassociated vector of anoperatorA correspondingto itseigenvalue λ
if for some m ≥ 1,we have

m
(A – λI) x ≠ 0, (A – λI) m+1 x = 0.
The number m is called the order of the associated vector x.

THEOREM. Let A be a linear operator in an n-dimensional unitary space V. Then there
m
is a basis {i } (k = 1, 2, ... , l, m = 1, 2, ... , n k , n 1 + n 2 + ··· + n l = n)in V consisting
k
of eigenvectors and associated vectors of the operator A such that the action of the operator
A is determined by the relations
1
Ai = λ k i 1 k (k = 1, 2, ... , l),
k
m
Ai m = λ k i + i m–1 (k = 1, 2, ... , l, m = 2, 3, ... , n k ).
k
k
k
1
Remark 1. The vectors i k (k = 1, 2, ... , l) are eigenvectors of the operator A corresponding to the
eigenvalues λ k.
m
Remark 2. The matrix A of the linear operator A in the basis {i k } has canonical Jordan form, and the
above theorem is also called the theorem on the reduction of a matrix to canonical Jordan form.

5.7. Bilinear and Quadratic Forms

5.7.1. Linear and Sesquilinear Forms

5.7.1-1. Linear forms in a unitary space.
A linear form or linear functional on V is a linear operator A in L(V, C), where C is the
complex plane.

THEOREM. For any linear form f in a finite-dimensional unitary space V,there is a
unique element h in V such that

f(x)= x ⋅ h for all x V.

Remark. This statement is true also for a Euclidean space V and a real-valued linear functional.

5.7.1-2. Sesquilinear forms in unitary space.
A sesquilinear form on a unitary space V is a complex-valued function B(x, y)oftwo
arguments x, y V such that for any x, y, z in V and any complex scalar λ, the following
relations hold:
1. B(x + y, z)= B(x, z)+ B(y, z).
2. B(x, y + z)= B(x, y)+ B(x, z).
3. B(λx, y)= λB(x, y).
¯
4. B(x, λy)= λB(x, y).