Generalisation of Polarization identity

In summary, the conversation discusses the relationship between a quadratic form on a vector space and its associated bilinear symmetric form. It is possible to generalize this to a k multilinear symmetric form, which is associated with an homogeneous polynomial of degree k. The conversation also mentions the use of the formal sum (x1 + x2 + ... + xk)^n, which can create a large sum with many binomial coefficients. A possible generalization is the polarization of an algebraic form.
  • #1
Geometry
6
2
Hello,

If I have a quadratic form ##q## on a ##\mathbb{R}## vectorial space ##E##, its associated bilinear symmetric form ##b## can be deduce by the following formula : ##b(., .) = \frac{q(. + .) - q(.) - q(.)}{2}##. So that, an homogeneous polynomial of degree 2 can be associated to a blinear symmetric form.

Can we generalize this at a k multilinear symmetric form? (I mean associate to an homogeneous polynomial of degree k a k multilinear symmetric form).

Have a nice day.
 
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  • #2
Their is a a way to developed the formal sum : ##(x_1 + x_2 + ... + x_k)^n## no? It might help but this will create a big sum with a lot of binomial coefficient.
 
  • #4
Hello,

This is very clear and very adapted for my needs. Thank you very much WWGD.

I wish you a good day.
 
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Related to Generalisation of Polarization identity

1. What is the Polarization Identity?

The Polarization Identity is a mathematical concept that relates the dot product of two vectors to their outer product. It states that the dot product of two vectors is equal to half the trace of their outer product. It is commonly used in linear algebra and quantum mechanics.

2. How is the Polarization Identity generalized?

The Polarization Identity can be generalized to higher dimensions by using matrices instead of vectors. In this case, the dot product becomes the matrix inner product and the outer product becomes the matrix outer product. The generalization allows the concept to be applied to a wider range of mathematical problems.

3. What is the significance of the Polarization Identity in quantum mechanics?

In quantum mechanics, the Polarization Identity is used to describe the state of a quantum system. It is used to calculate the probability of a measurement outcome and to determine the evolution of a quantum system over time. The identity is also used in the mathematical formulation of quantum mechanics, known as the bra-ket notation.

4. How is the Polarization Identity related to the Law of Cosines?

The Polarization Identity is closely related to the Law of Cosines, which states that the square of the length of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the angle between them. This relationship can be seen when the Polarization Identity is applied to vectors in two or three dimensions.

5. Can the Polarization Identity be used in other areas of science?

Yes, the Polarization Identity has applications in various areas of science, including physics, engineering, and computer science. It is used in signal processing, image processing, and pattern recognition. It is also used in the study of electromagnetic waves and polarization of light.

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