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URL: https://nakafa.com/en/subjects/ai-ds/linear-methods/scalar-product
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Understand scalar products in vector spaces: bilinear forms, positive definite matrices, Hermitian properties, and real vs complex implementations.
---
## Definition of Scalar Product
A scalar product is a fundamental operation that allows us to compute the "multiplication" between two vectors with the result being a scalar number. Through every positive definite matrix, we can define a scalar product on vector spaces.
A scalar product on complex vector space $$\mathbb{C}^n$$ is a sesquilinear Hermitian form that is positive definite with three main properties.
Visible text: A scalar product on complex vector space is a sesquilinear Hermitian form that is positive definite with three main properties.
### Sesquilinear Property
For all vectors $$x, y, z \in \mathbb{C}^n$$ and scalars $$\lambda, \mu \in \mathbb{C}$$, the scalar product satisfies:
Visible text: For all vectors and scalars , the scalar product satisfies:
Component: MathContainer
Children:
```math
\langle \lambda x + \mu y, z \rangle_A = \overline{\lambda} \langle x, z \rangle_A + \overline{\mu} \langle y, z \rangle_A
```
```math
\langle x, \lambda y + \mu z \rangle_A = \lambda \langle x, y \rangle_A + \mu \langle x, z \rangle_A
```
This property is called sesquilinear because it is semilinear in the first argument and linear in the second argument.
### Hermitian Property
For all vectors $$x, y \in \mathbb{C}^n$$, the following holds:
Visible text: For all vectors , the following holds:
```math
\langle y, x \rangle_A = \overline{\langle x, y \rangle_A}
```
### Positive Definite Property
For all vectors $$x \in \mathbb{C}^n$$, the following holds:
Visible text: For all vectors , the following holds:
Component: MathContainer
Children:
```math
\langle x, x \rangle_A \geq 0
```
```math
\langle x, x \rangle_A = 0 \Leftrightarrow x = 0
```
Thus, a scalar product is a sesquilinear Hermitian form that is positive definite on $$\mathbb{C}^n$$.
Visible text: Thus, a scalar product is a sesquilinear Hermitian form that is positive definite on .
## Scalar Product on Real Space
If $$A \in \mathbb{R}^{n \times n}$$ is a positive definite matrix, then the mapping:
Visible text: If is a positive definite matrix, then the mapping:
```math
\langle \cdot, \cdot \rangle_A : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} : (x, y) \mapsto \langle x, y \rangle_A = x^T A y
```
has the following properties:
1. **Bilinear**: For all vectors $$x, y, z \in \mathbb{R}^n$$ and scalars $$\lambda, \mu \in \mathbb{R}$$, the following holds:
```math
\langle \lambda x + \mu y, z \rangle_A = \lambda \langle x, z \rangle_A + \mu \langle y, z \rangle_A
```
```math
\langle x, \lambda y + \mu z \rangle_A = \lambda \langle x, y \rangle_A + \mu \langle x, z \rangle_A
```
This property is called bilinear because it is linear in both arguments.
2. **Symmetric**: For all vectors $$x, y \in \mathbb{R}^n$$, the following holds:
```math
\langle y, x \rangle_A = \langle x, y \rangle_A
```
3. **Positive Definite**: For all vectors $$x \in \mathbb{R}^n$$, the following holds:
```math
\langle x, x \rangle_A \geq 0
```
```math
\langle x, x \rangle_A = 0 \Leftrightarrow x = 0
```
Thus, a scalar product is a symmetric bilinear form that is positive definite on $$\mathbb{R}^n$$.
Visible text: 1. **Bilinear**: For all vectors and scalars , the following holds:
This property is called bilinear because it is linear in both arguments.
2. **Symmetric**: For all vectors , the following holds:
3. **Positive Definite**: For all vectors , the following holds:
Thus, a scalar product is a symmetric bilinear form that is positive definite on .
## Scalar Product on Complex Space
Generalization to complex space involves the important role of complex conjugate. If $$A \in \mathbb{C}^{n \times n}$$ is a positive definite matrix, then the mapping:
Visible text: Generalization to complex space involves the important role of complex conjugate. If is a positive definite matrix, then the mapping:
```math
\langle \cdot, \cdot \rangle_A : \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C} : (x, y) \mapsto \langle x, y \rangle_A = x^H A y
```
has the same properties as the general definition above, namely sesquilinear in the first argument, linear in the second argument, Hermitian with complex conjugate, and positive definite.
This shows that when we move to complex space, the complex conjugate plays an important role in maintaining a consistent scalar product structure.
## Matrix Representation
An important result in scalar product theory is that every scalar product can be expressed in the form of an appropriate matrix.
Every scalar product $$s(x, y)$$ on $$\mathbb{R}^n$$ as well as on $$\mathbb{C}^n$$ can be expressed in the form:
Visible text: Every scalar product on as well as on can be expressed in the form:
Component: MathContainer
Children:
```math
s(x, y) = x^T A y \text{ (for real case)}
```
```math
s(x, y) = x^H A y \text{ (for complex case)}
```
through an appropriate positive definite matrix $$A \in \mathbb{R}^{n \times n}$$ or $$A \in \mathbb{C}^{n \times n}$$.
Visible text: through an appropriate positive definite matrix or .
The elements of matrix $$A$$ can be computed as:
Visible text: The elements of matrix can be computed as:
```math
a_{ij} = s(e_i, e_j) \text{ for } i, j = 1, \ldots, n
```
where $$e_i$$ is the standard basis vector.
Visible text: where is the standard basis vector.
## Example of Standard Scalar Product
The simplest example is the standard scalar product that uses the identity matrix $$I$$.
Visible text: The simplest example is the standard scalar product that uses the identity matrix .
The standard scalar product is defined as:
Component: MathContainer
Children:
```math
\langle x, y \rangle = x^T y \text{ (for real case)}
```
```math
\langle x, y \rangle = x^H y \text{ (for complex case)}
```
This standard scalar product is obtained by using the identity matrix as its representation matrix, so that all required properties are satisfied naturally.
> Through the relationship between scalar products and positive definite matrices, we can construct various types of scalar products that suit specific application needs.