A function f(𝐱) on ℝ^n is called homogeneous of degree k if f(c 𝐱)=c^k f(𝐱) for all scalars c. (

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A function f(𝐱) on ℝ^n is called homogeneous of degree k if...

A function $f(\mathbf{x})$ on $\mathbb{R}^n$ is called homogeneous of degree $k$ if $f(c \mathbf{x})=c^k f(\mathbf{x})$ for all scalars c. (a) Given $\mathbf{a} \in \mathbb{R}^n$, show that the linear form $\ell(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}=a_1 x_1+\cdots+a_n x_n$ is homogeneous of degree 1 . (b) Show that the quadratic form $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}=\sum_{i, j=1}^n k_{i j} x_i x_j$ is homogeneous of degree 2. (c) Find a homogeneous function of degree 2 on $\mathbb{R}^2$ that is not a quadratic form.

A function $f(\mathbf{x})$ on $\mathbb{R}^n$ is called homogeneous of degree $k$ if $f(c \mathbf{x})=c^k f(\mathbf{x})$ for all scalars c. (a) Given $\mathbf{a} \in \mathbb{R}^n$, show that the linear form $\ell(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}=a_1 x_1+\cdots+a_n x_n$ is homogeneous of degree 1 . (b) Show that the quadratic form $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}=\sum_{i, j=1}^n k_{i j} x_i x_j$ is homogeneous of degree 2.
(c) Find a homogeneous function of degree 2 on $\mathbb{R}^2$ that is not a quadratic form.

Key Concepts

Homogeneous Functions

A homogeneous function is one for which scaling all the inputs by a common factor results in the output being scaled by some power of that factor. Mathematically, this means a function f on R^n satisfies f(c * x) = c^k f(x) for all scalars c, where k is called the degree of homogeneity.

Degree of Homogeneity

The degree of homogeneity, denoted by k in the relation f(c * x) = c^k f(x), indicates how the function scales with respect to its inputs. It is a key concept in understanding the behavior of functions under scaling transformations and plays an important role in areas such as differential equations, optimization, and economic theory.

Linear Forms

A linear form is a function defined by the dot product between a fixed vector and the variable vector, typically expressed as a1 x1 + a2 x2 + ... + an xn. Because it meets the criterion f(c * x) = c f(x), linear forms are homogeneous functions of degree 1, highlighting their direct proportionality with input changes.

Quadratic Forms

Quadratic forms are functions that involve the product of pairs of variables, usually expressed as x^T K x where K is a matrix. Due to the fact that each term in the sum involves the product of two components of x, quadratic forms satisfy f(c * x) = c^2 f(x), making them homogeneous of degree 2. This scaling property is critical in analyzing conic sections, optimization problems, and the geometry of surfaces.

Non-Quadratic Homogeneous Functions

Not all homogeneous functions of a given degree are quadratic forms. There exist functions that, despite being homogeneous of degree 2 (or any other degree), do not have the structure of a polynomial of total degree 2. Exploring such functions broadens the understanding of homogeneity beyond classical polynomial forms, illustrating the diversity of functions with scaling properties.

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