Bilinear and Quadratic Forms
The theory of bilinear form and quadratic form is used [5 ] in the analytic geometry for getting the classification of the conics and of the quadrics.
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Bilinear and Quadratic Forms
5.1 Bilinear and Quadratic Forms The theory of bilinear form and quadratic form is used [5] in the analytic geometry for getting the classification of the conics and of the quadrics. It is also used in physics, in particular to describe physical systems subject to small vibrations. The coefficients of a bilinear form one behave to certain transformations like the tensors coordinates. Tensors are useful in theory of elasticity (the deformation of an elastic medium is described through the deformation tensor). Definition 5.1 (see [1], p. 150). A mapping b : V × V → K is called a bilinear form on V if it satisfies the conditions: 1. b (αx + β y, z) = αb (x, z) + βb (y, z) , (∀) α, β ∈ K , (∀) x, y, z ∈ V, 2. b (x, α y + βz) = αb (x, y) + βb (x, z) , (∀) α, β ∈ K , (∀) x, y, z ∈ V. Definition 5.2 (see [1], p. 150). We say that the bilinear form b : V × V → K is symmetric (antisymmetric) if b (x, y) = b (y, x) (respectively, b (x, y) = −b (y, x). Consequences 5.3 (see [2], p. 116). If the mapping b : V × V → K is a bilinear form then: x = b x, 0 = x∈V (1) b 0, 0, (∀) n n (2) (a) b α (i) x i , y = α (i) b (x i , y) , i=1
i=1
(1) , . . . , α (n) ∈ K , (∀) x , . . . , x , y ∈ V (∀) α 1 n n n (i) β yi = b x, y i , (b) b x, i=1
i=1
(∀) β (1) , . . . , β (n) ∈ K , (∀) x, y 1 , . . . , y n ∈ V.
G. A. Anastassiou and I. F. Iatan, Intelligent Routines II, Intelligent Systems Reference Library 58, DOI: 10.1007/978-3-319-01967-3_5, © Springer International Publishing Switzerland 2014
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5 Bilinear and Quadratic Forms
Definition 5.4 (see [1], p. 150). If b : V × V → K is a symmetric bilinear form, the mapping f : V → K , defined by f (x) = b (x, x), for all x ∈ V is called the quadratic form associated to b. Remark 5.5 (see [1], p. 150). Knowing the quadratic form f, allows to get the symmetric bilinear form, associated to f as: b (x, y) =
1 [ f (x + y) − f (x) − f (y)] , (∀) x, y ∈ V 2
(5.1)
Definition 5.6 (see [1], p. 150). The symmetric bilinear form b associated to the quadratic form f is called the polar form of the quadratic form f . Example 5.7 (see [3], p. 93). The quadratic form corresponding to the real scalar product (which is a symmetric bilinear form) is the square of the Euclidean norm: f (x) =< x, x >= x2 , (∀) x ∈ V. Let V be an n finite dimensional vector space over K , n ≥ 1 and B = {a 1 , . . . , a n } one of its basis. If b : V × V → K is a bilinear form then (∀) x, y ∈ V it results: x=
n
x (i) a i , y =
i=1
therefore b (x, y) =
n
y ( j) a j ;
j=1
n n
ai j x (i) y ( j) ,
(5.2)
i=1 j=1
where
ai j = b a i , a j , (∀) i, j = 1, n.
The expression (5.2) constitutes [1] the analytic expression of the bilinear form b relative to the basis B, and A ∈Mn (K ), A = ai j 1≤i, j≤n represents the associated matrix of the bilinear form b relative to the basis B. From (5.2) one obtains [1] the analytic expression of the bilinear form f : V → K relative to the basis B of V : n n n f (x) = ai j x (i) y ( j) , (∀) x = x
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