Quadratic Forms With Applications

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, . . . , x_n is a homogeneous polynomial of degree 2 in K[x₁, . . . , x_n], 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) = X^T 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 ) = X^T 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

1. Given a  K-vector space V , the scalar multiplication definedfrom K V

into V as

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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 f^j(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 H_f (x₀) and called the Hessian of f at x₀ such that

f^jj(x₀)(u, v)  =  (u, H_f (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,

Bibliography

• Lang, S. : Calculus of several variables. 2 ^nd Springer New-York. 1987.

• Jost, J. : Postmodern Analysis. Springer Berlin.
• Berkovitz,D. : Convexity and Optimization in Rn. John Wiley & Sons, Inc. 2001.
• Pfister, A. : Quadratic Forms with Applications to AlgebraicGeometry and Cambridge University Press. 1995.
• Gregory, : Quadratic Form Theory and Differential Equations. Math- ematics in Sciences and Engineering 152. Academic Press 1980.
• Lax, : Functional Analysis. Wiley-Interscience2002.
• O’Meara, O.T. : Introduction to Quadratic 3rd Ed. Springer Berlin1973.
• Simon, : Hamiltonians defined as quadratic forms. Comm. Math. Phys. 21. (1971), 192-210.

• Troutman, L. : Variational Calculus and Optimal Control. Optimiza- tion with elementary convexity. 2nd Ed. Springer1996
• Simon, B. : A canonical decomposition of quadratic forms with applica- tions to monotone operators. Journal of Functional Analysis 28. (1978) 337-385.

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