Does the following property have a name?

Question

I came up with this thing that takes in two vectors and returns another vector. As of right now it only works for Euclidean vectors that are greater than one dimensional. I'm not going to go over what this thing actually does, it's pretty simple anyway and I don't think it's very important to my question.

All I want to know is that this thing has the property that I if I scale BOTH vectors by the same constant and then apply this transformation, it is the same as applying the transformation first and then scaling the result by that constant. To be clear, this is not one of the properties of a Bilinear map because both vectors are being scaled by the same constant at the same time.

Written out it looks like this:

ε(a∙v,a∙w)=a∙ε(v,w)

Where the epsilon represents the transformation, v and w are the two euclidean vectors, and the a is want is scaling each vector (I couldn't figure out how to get the vector arrows).

What I want to know is if this property has a name or if anyone knows of a transformation that has this property.

Thanks.

Answer 1

This is called being homogeneous of degree $1$. More generally, here's a definition:

Let $k \geq 0$ be an integer, and let $V, W$ be vector spaces over a field $\Bbb{F}$ (if you wish, you may suppose the field is $\Bbb{R}$). A function $f: V \to W$ is called homogeneous of degree $k$ if for all $\xi \in V$ and all $a \in \Bbb{F}$, we have \begin{align} f(a \xi) &= a^k f(\xi). \end{align}

In the case you were looking at, you haven't fully made all the details explicit, but I assume you had something like this in mind: you have three Cartesian spaces $\Bbb{R}^m, \Bbb{R}^n$ and $\Bbb{R}^p$ and a function $\epsilon : \Bbb{R}^m \times \Bbb{R}^n \to \Bbb{R}^p$, such that for all $v \in \Bbb{R}^m, w \in \Bbb{R}^n$ and all $a \in \Bbb{R}$, we have \begin{align} \epsilon(av,aw) = a \cdot\epsilon(v,w). \end{align}

Well, this is also encompassed in the definition I gave above. Take the field $\Bbb{F}$ to be $\Bbb{R}$, take $V = \Bbb{R}^m \times \Bbb{R}^n$ and $W = \Bbb{R}^p$, and the degree of homogeneity $k$ to be $1$.

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Examples of this include:

• The identity map on any vector space is homogeneous of degree $1$.
• Any "linear polynomial of $n$-variables" like $f: \Bbb{R}^n \to \Bbb{R}$ \begin{align} f(x_1, \dots, x_n) = \sum_{i=1}^n c_i x_i \end{align} is homogeneous of degree $1$.
• For fun, any norm on a vector space can be used to define a homogeneous function of degree $2$, as follows: $f: V \to \Bbb{R}$ defined as $f(\xi) := \lVert \xi \rVert^2$.
• More generally, given any $k$-times multilinear function $F: \underbrace{V \times \dots V}_{k \text{ times}} \to W$, consider the function $f: V \to W$ defined as \begin{align} f(\xi) &:= F(\xi, \dots, \xi). \end{align} This will define a homogeneous function of degree $k$.