Bilinear and Quadratic forms are linear transformations in more than one variable over a vector space. A homogeneous polynomial in one, two, or n variables is called form. Forms are classified in two ways. According to the number of variables, they are called unary, binary, ternary, etc. According to the degree, they are classified as linear, quadratic, cubic, etc.

Thus, bilinear form is a polynomial function of two variables which is linear in both the variables. The quadratic form is a particular case of the bilinear form, as the two sets of variables used in the bilinear form are equal.

Bilinear Form

A polynomial of sets of variables (x₁, x₂, x₃, …, x_m) and (y₁, y₂, y₃, …, y_n) in which the coefficients are numbers and are homogenous, that is, each term of the polynomial have the same degree. Also, the polynomial is linear in both variables. Hence, the definition of the bilinear form as linear transformation over vector spaces is given as —

If V is a vector space of finite-dimensional over the field F, then a bilinear form on V is a mapping f: V × V → F, which satisfies:

i) f(ax₁ + bx₂, y) = a f(x₁, y) + b f(x₂, y)

That means, f is linear as the first variable.

ii) f(x, ay₁ + by₂) = a f(x, y₁) + b f(x, y₂)

That means, f is linear as a second variable.

For every x_i, y_i in V and a, b in F.

For example, the zero function from V × V into F, that is, 0: V × V → F we shall show that this function satisfies the above two conditions

Now 0(ax₁ + bx₂, y) = 0 = 0 + 0 = a.0 + b.0 = a 0(x₁, y) + b 0(x₂, y)

and 0(x, ay₁ + by₂) = 0 = 0 + 0 = a.0 + b.0 = a 0(x, y₁) + b 0(x, y₂)

for every x_i, y_i in V and a, b in F.

Thus, the zero function is a bilinear form.

Notice, the difference between linear and bilinear form is f(x, y) = x + y is linear whereas as

f(x, y) = xy is bilinear.

Thus, in a more general sense, f(x, y) = a.xy for any scalar a is bilinear.

Quadratic Form

The quadratic form is a particular case of a bilinear form. The two sets of variables in bilinear form (x₁, x₂, x₃, …, x_m) and (y₁, y₂, y₃, …, y_n) become quadratic form if the two sets are equal and x_i = y_i for each i. For example, f(x,y) = x² – 2y² + 5xy is a real quadratic form in two variables x and y.

Now, quadratic form as linear transformation over a vector space V is defined as if f is a symmetric bilinear form, that is f(x, y) = f(y, x) for every x, y in V, the quadratic form is given by q(x) = f(x, x) for every x in V.

Here, q has the property q(ax) = a² q(x) where a is a scalar and x is in V. As, q(ax) = f(ax, ax) = a² x² = a q(x).

For example, let us take a bilinear for f defined by matrix

Then its quadratic form will be

= 2x² + 5y²

Properties of Bilinear and Quadratic Forms

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

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

(ii) Q(0) = 0 for 0 in V

(iii) Q(-x) = Q(x) for x in V

• If Q is a quadratic form on V, then Q(x + y) = (x + y)² = Q(x) + 2f(x, y) + Q(y) where f is bilinear form of x and y. Thus f(x, y) = ½[ Q(x + y) – Q(x) – Q(y)]
• The bilinear form of (x, y) and (w, z) in V × V in terms of quadratic form is given as

f((x, y), (w, z)) = ½[Q(x + w, y + z) – Q((x, w)) – Q((y, z))]

Related Articles

• Vector Space
• Eigen Values
• Matrix and Determinants
• Rank of Matrix
• Special Matrices

Solved Examples on Bilinear and Quadratic Forms

Example 1:

Let 𝛼 = (x₁, x₂, x₃) and 𝛽 = (y₁, y₂, y₃) and the bilinear form of 𝛼 and 𝛽 is given as

f(𝛼, 𝛽) = x₁ y₁ + 2 x₁ y₂ + 5 x₁ y₃ – 2 x₂ y₁ + x₂ y₃ – 6 x₃ y₂ + 6 x₃ y₃. Find the matrix of f and rank of f.

Solution:

We have

f(𝛼, 𝛽) = x₁ (y₁ + 2 y₂ + 5 y_3,) + x₂ (– 2 y₁ + 0 y₂ + y₃) + x₃(0 y₁ – 6 y₂ + 6 y₃)

=

=

Thus matrix of f is

Since the determinant of above matrix is non-zero, hence its rank is 3

Thus, rank of f is also 3.

Example 2:

Find the quadratic form of the symmetric matrix A =

Solution:

Let 𝛼 = (x, y)

Then the quadratic form Q(𝛼) of A is given by

Q(𝛼) =

= 2x² – 6xy + 3y²

Thus, 2x² – 6xy + 3y² is the quadratic form of the given matrix.