The bilinear and the quadratic form is an important concept in linear algebra and geometry that is widely used in various areas of applied mathematics and science, including physics, functional analysis, and control theory. This article will give you a basic introduction about what a bilinear and quadratic form is, and how both are applied in a wider mathematical context.

What's a Bilinear form?

Simply put, the bilinear form is the function of two vector variables that produce real (or complex) numbers and is linear to each of the variables. bilinear forms are usually defined above vector space and can be described as function:

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with VVV is a vector space, and B (u, v) B (u) B (u, v) is a real number generated from two uuu and vvv vv vvv v. BBB function is called bilinear because it's linear against both arguments:

1. B (u1 + u2, v) = B (u1, v)
2. B (u, v1 + v2) = B (u, v1) + B
3. B

One of the most common examples of bilinear forms is product in (inner product) in Euclidean space, which is a symmetrical bilinear form:

You can't do that.

In this context, the bilinear forms describe operations related to long, angle and orthongonality in the vector room.

What's a quadratic form?

The quadratic form is a special case of bilinear forms where both arguments in bilinear functions are the same vector. Mathematically, QQQ quadratic form above VVV vector space is a function of one vector variable defined as:

Q (u) = B (u, u) Q (u) = B (u) Q (u)

where BBB is a bilinear form. The quadratic form is often used to describe quantity such as vector length or energy in a physics system. For example, in Euclidean space, the quadratic form is associated with the norm or the length of a vector:

Q

In the second dimension, a quadratic form also appears in circular equations or ellipses:

x2 + y2 = r2 (circle) x ^ 2 + y ^ 2 = r

Bikinear and Kuadratic Forms Applied

1. Physical and Mechanic
   The quadratic form is often used in physics, especially in kinetic energy calculations or potential energy in mechanics. The kinetic energy of a moving object, for example, is a quadratic form at vector speed: Ek = 12mv2E _ k = frac {1} {2} m v ^ 2Ek = 21 mv2 In classical mechanics, stable systems are usually associated with the form of a positive-definite quadratic, which is always a positive form of a squared except for vector zero.
2. Quadratic optimization
   In applied computer science and math, the quadratic form is widely used in optimizations, especially in quadratic programming. Quadratic optimizations are looking for a minimum or maximum value of a quadratic function, which can be used in various applications from the economy to automatic control.
3. Diferential Geometry
   In geometry, the bilinear form is used to define concepts like curvature and metric in differential varieties. The bilinear form also helps define the tensors used in general relativity theory to describe space-time.
4. Matrique Theory
   bilinear and quadratic forms can be expressed in matrix form. If we take the AAA symmetrical matrix, then the quadratic form can be written as: Q (x) = xtaxQ (x) = x ^ T A xQ (x) = xtaxx here, xxx is the column vector, AAA is the symmetrical matrix, and xTx ^ TxT is the transmitter. The presentation in the form of this matrix is very important in numerical analysis and scientific computation, where quadratic form calculations are often done to evaluate the stability of the system or to solve optimizations problems.
5. Group Theory and Representation
   The bilinear form also appears in the group theory representation, where they're used to describe the algebraic structure of the vector space operated by the group.

Conclusion

The bilinear and quadratical form is a fundamental concept of mathematics that has various applications in physics, computers and engineering. The bilinear form describes a linear relationship between two vectors, while the quadratic form is a special form of bilinear that measures the length or energy of a vector. The application of this concept can be seen in various disciplines, including mechanics, optimization and geometry.

source: Axler, S. 2015. Linear Algebra Done Right. Springer.