Bilinear and Quadratic Forms in Linear Algebra

In mathematics, bilinear and quadratic forms involve the linear transformation of several variables defined on a vector space V.

• Form refers to a homogeneous polynomial involving one, two, or n variables. 
• They are categorized as unary, binary, ternary, etc., according to the number of variables.
• The bilinear form is a type of polynomial function of two variables that is linear in both variables.
• The most common example of this form involves a dot product in an Euclidean space.
• Quadratic form, on the other hand, is a special instance of bilinear form where two sets of variables in the bilinear form are equal.

Key Terms: Bilinear and Quadratic Forms, Linear Algebra, Bilinear Form, Quadratic Form, Polynomial, Function, Variables, Vector Space

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Bilinear Form

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The bilinear form is a type of bilinear map that involves more than one variable where each set of variables is linear. 

• It is a type of polynomial function where each variable has the same degree, and the coefficients are numbers.
• A bilinear form on a vector space V, which is finite-dimensional over a field F, is a mapping f: V × V → F that meets the following conditions:
  i) f(ax₁ + bx₂, y) = a f(x₁, y) + b f(x₂, y), indicating that the first variable, f, is linear.
  ​ii) f(x, ay₁ + by₂) = a f(x, y₁) + b f(x, y₂), indicating f is a linear function as a second variable.
• The distinction between the bilinear and linear forms is that, whereas f(x, y) = x + y is linear, xy = f(x, y) is bilinear. 
• For every scalar a, f(x, y) = a.xy is bilinear in the more generic sense.

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Quadratic Form

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Quadratic forms are homogeneous quadratic polynomials that are considered a particular case of bilinear form. 

• The two variables in bilinear form (x₁, x₂, x₃, …, x_m) and (y₁, y₂, y₃, …, y_n) are transformed into quadratic form if the two variables are equal for each x_i = y_i.
• In vector space V, if f is a symmetric bilinear form where f(x, y) = f(y, x) for every x, y in vector space V, then a quadratic form is given by q(x) = f(x, x) for every x in V.
• For this case, q satisfies the condition q(ax) = a² q(x) and consequently, q(ax) = f(ax, ax) = a² x² = a q(x).
• It is used in number theory, linear algebra, group theory, differential geometry and the second fundamental form.

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Properties of Bilinear and Quadratic Forms

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Some important properties regarding bilinear and quadratic forms are as follows:

• Consider two bilinear forms, f and g, on a dimensional space V(F) where the sum of two bilinear forms f and g is also a bilinear form, defined by (f + g)(x, y) = f(x, y) + g(x, y).
• Let c be a scalar function, and f be in the bilinear form on V × V. Then, for each x, y in V, (cf)(x, y) = c f(x, y) defines cf as a bilinear form on V × V.
• The rank of a function in bilinear form f on a vector space V indicates the rank of the matrix representation of f.
• Given a vector space V and a quadratic form Q, then

(i) Q(ax) = a² Q(x) , where x is in V and an is any scalar

(ii) When 0 in V, Q(0) = 0.

(iii) For each x in V, Q(-x) = Q(x).

• In terms of quadratic form, the bilinear form of (x, y), (w, z) in V × V is as follows: f((x, y), (w, z)) = ½[Q(x + w, y + z) – Q((x, w)) – Q((y, z))]
• If Q is in quadratic form, then Q(x + y) = (x + y)² = Q(x) + 2f(x, y) + Q(y), where f is the bilinear form of x and y. Consequently, f(x, y) = ½Q(x + y)-Q(x)-Q(y)]

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Things to Remember

• In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V, whose components are vectors, over a field K, whose elements are scalars.
• Homogeneous quadratic polynomials in n variables are known as quadratic forms.
• Modules over a ring can be included in the formulation of a bilinear form by substituting module homomorphisms for linear mappings.
• Numerous number theory topics, including the theory of quadratic fields, continuing fractions, and modular forms, have included a great deal of research on binary quadratic forms.
• A non-zero quadratic form in n variables defines a (n − 2) dimensional quadric in the (n − 1) dimensional projective space using homogeneous coordinates.

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