definition of quadratic form by \(2\)-symmetric-tensor
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of quadratic form by \(2\)-symmetric-tensor.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\( \Sigma_2 (V: W)\): \(= \text{ the symmetric-tensors space with respect to } F \text{ and } 2 \text{ same vectors spaces and vectors space over } F\)
\( t\): \(\in \Sigma_2 (V: W)\)
\(*\widetilde{t}\): \(: V \to W, v \mapsto t (v, v)\)
//
Conditions:
//
2: Note
Usually, a "quadratic form" is \(\widetilde{M}: \mathbb{R}^d \to \mathbb{R}, v \mapsto v^t M v\), where \(M\) is a \(d \times d\) real symmetric matrix, which is really the components expression with respect to any basis for \(V\) of a special case of this definition such that \(F = \mathbb{R}\), \(V\) is any \(d\)-dimensional real vectors space, \(W = \mathbb{R}\), the components of \(t\) are \(M\), and the components of \(v\) is \((v^1, ..., v^d)\): \(t (v, v) = M_{j, l} v^j v^l\), which is conveniently written as \(M^j_l v_j v^l = v^t M v\).
\(t\) determines \(\widetilde{t}\), and in fact, \(\widetilde{t}\) determines \(t\), by the proposition that the quadratic form by any \(2\)-symmetric-tensor determines the tensor.
