Quadratic Forms With Applications
Chapter One
PREAMBLE OF THE STUDY
A quadratic form over a field K (K = R or K = C) in finitely many indeterminate x1, . . . , xn is a homogeneous polynomial of degree 2 in K[x1, . . . , xn], unless it is identically zero. The main property of real quadratic forms in the finite dimensional case is that every real quadratic form is orthogonally sim- ilar (i.e., can be transformed by an orthogonal change of the indeterminates considered as coordinates) to a quadratic form which is the sum of multiples of squares of the indeterminates [16].
In fact a quadratic form in a finite set of indeterminates over K, as a homoge- neous quadratic polynomial in the indeterminates with coefficients in K, can be studied by means of matrices because any such a quadratic form Q can be expressed as Q(X) = XT AX, where X is a column vector with the indetermi- nates as elements and A a symmetric matrix over K. Thus it is the quadratic form associated with the symmetric bilinear form defined from Kn Kn to K by f (X, Y ) = XT AY ; X, Y ∈ Kn , and this gives rise to a duality.
Chapter Two
Bilinear Maps and Forms
Bilinear maps
Definition 2.1.1 (Bilinear maps)
Let E, F and G be three arbitrary vector spaces over K.
A bilinear map Φ from E F into G is a mapping Φ : E F G satisfying the following two conditions :
- Φ( α1x1+ α2x2, y) = α1Φ(x1 , y) + α2Φ(x2 , y) for all x1, x2 ∈ E, y ∈ F and α1, α2 ∈ K.
- Φ( x, α1y1+α2y2) = α1Φ(x, y1) + α2Φ(x, y2) for all x ∈ E, y1, y2 ∈
F and α1, α2 ∈ K.
This means that Φ is separately linear with respect to each of its two argu- ments (variables).
When E = F , a bilinear map from E2 = E E into G is called a G-valued
bilinear map on E.
Remark 2.1.2
Note that the above two conditions that define the bilinearity of Φ are also respectively equivalent to the following :
Φ( x1+ αx2, y) = Φ(x1 , y) + αΦ(x2 , y) for all x1, x2 ∈ E, y ∈ F and α ∈ K.
Φ( x, y1+ αy2) = Φ(x, y1) + αΦ(x, y2) for all x ∈ E, y1, y2 ∈ F and α ∈ K.
There are many interesting bilinear maps in the literature. Let’s us mention few ones.
Examples 2.1.3
- Given a K-vector space V , the scalar multiplication definedfrom K V
into V as
Chapter Three
Quadratic forms
Generalities on Quadratic Forms and Spaces
Definition 3.1.1
A quadratic form on a K-vector space V , is a functional Q on V such that there exists a bilinear form f on V satisfying
Q(x) = f (x, x) , ∀ x ∈ V .
First Properties
Proposition 3.1.2 (Polar form of a quadratic form)
For every quadratic form Q on a K-vector space V , there exists a unique symmetric bilinear form ϕ on V such that
Q(x) = ϕ(x, x) , ∀x ∈ V .
This unique symmetric bilinear form ϕ corresponding to Q is called the polar form of Q and can be expressed by
ϕ(x, y) = 1 Q(x + y) Q(x) Q(y) , x, y V .
Consequently, there is a one-to-one correspondance between the class of quadratic forms of a vector space V and the class of symmetric bilinear form on V .
Proof.
Q being a quadratic form, there exists a bilinear form f such that
Q(x) = f (x, x) for all x ∈ V .
Thus it is not hard to check that ϕ = f∗ (the symmetric part of f , cf. Definition
…) is the unique symmetric bilinear form such that Q(x) = ϕ(x, x) for all
Chapter Four
Applications
Quadratic forms and Unconstrained Opti- mization
Proposition 4.1.1 [23],[19]
Let H be a real Hilbert, Ω be a nonempty open set of H and f : Ω → R be a function. Let x0 ∈ Ω.
- Iff is differentiable at x0, then the derivative (in the sense of Fr´echet) of f at x0 is a bounded linear functional on H and so there exists a unique vector denoted by ∇f (x0) and called the gradient of f at x0 such that fj(x0)(h) = (∇f (x0), h) , ∀h ∈ H .
- If f is of class C2, then the second order derivative (in the sense of Fr´echet)of f at x0 is a symmetric bounded bilinear form on H and so there exists a unique bounded symmetric operator denoted by Hf (x0) and called the Hessian of f at x0 such that
fjj(x0)(u, v) = (u, Hf (x0)v) , ∀u, v ∈ H .
Theorem 4.1.2 (Optimality Necessary Condition)[3]
Let Ω be a nonempty open set in Rn, let f be a real-valued function defined on Ω and suppose that x0 ∈ Ω is a local minimizer.
If f has first order partial derivatives at x0, then
∂f
(x0) = 0 , for all i = 1, 2, . . . , n.
∂x
In particular, if f is differentiable at x0, then x0 is a critical point of f ; that is,
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