Exploring Functions and Matrices with Homogeneous Properties

In this article, we will delve into the exploration of functions and matrices that exhibit specific homogeneous properties. We will start by discussing non-constant non-diagonal matrices and connections to exponential functions. Additionally, we will explore trivial solutions and the logarithmic approach to finding such functions.

Non-Constant Non-Diagonal Matrices

To extend on the other answers, we present a solution involving non-constant, non-diagonal matrices. Instead of using a one-dimensional exponential function, we can create a two-dimensional solution by using a matrix exponentiation method.

Consider the function ( f(x) e^{xA} ), where ( A ) is a 2x2 matrix. If ( A ) is a diagonal matrix, we obtain the standard diagonal solution with exponentials. However, for a non-diagonal matrix like

( A  begin{pmatrix} 0  -theta  theta  0 end{pmatrix} )

We get the following:

( f(x)  begin{pmatrix} cos(theta x)  -sin(theta x)  sin(theta x)  cos(theta x) end{pmatrix} )

Trivial Solutions

Aside from non-diagonal matrices, there are also trivial solutions. These include the zero matrix:

( f(x)  0_{2times2} )

and the identity matrix:

( f(x)  I_{2times2} )

For the general case, any constant matrix that is its own square will satisfy the equation. If your primary interest is in proving existence, you don't need to look for a complete solution. You can take the straightforward approach by choosing ( A ) as a constant matrix.

Logarithmic Approach

For a more fundamental understanding, consider the simplified problem of a scalar-valued function ( g(x) ) with the property ( g(x)^2 g(2x) ). Taking the natural logarithm on both sides, we get:

( 2 log(g(x))  log(g(2x)) )

This equation suggests that ( g(x) ) must be a homogeneous function. For instance, if we take ( h(x) log(g(x)) ), then ( h(x) ) must be proportional to ( x ). A logical choice would be:

( h(x)  cx ) where ( c ) is a constant.

Now, the corresponding exponential function gives us:

( g(x)  e^{cx} )

To extend this to a matrix, we can define a diagonal matrix ( A ) with the form of ( g(x) ) along the diagonal. Thus, the matrix ( f(x) ) becomes:

( f(x)  e^{xA} )

where ( A ) is a diagonal matrix with entries ( cx ).