Derivative of a projective transformation

Question

Assume $A$ is a matrix from $R^{n\times n}$, $A:R^n\rightarrow R^n$. Then $A$ induces a projective transformation $f:RP^{n-1}\rightarrow RP^{n-1}$. For example, $\\$

$$\begin{pmatrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix}\cdot\begin{pmatrix} 1 \\ x \\ y \end{pmatrix}=\begin{pmatrix} 4 \\ 3x \\ 2y \end{pmatrix} \sim \begin{pmatrix} 1 \\ 3x/4\\ y/2 \end{pmatrix} $$ So $f$, induced by $diag(4\quad 3\quad 2)$, sends $(1\quad x\quad y)^T$ to $(1\quad 3x/4 \quad y/2)^T$. Similarly it sends $(0\quad 1\quad y)^T$ to $(0 \quad 1\quad 2y/3)^T$. The question is, how to define the derivative of such a transformation? I think addition is not well-defined in projective space?

Background:

(i) I need to compute $\int \delta(x-f(x))g(x)dx$, where $\delta$ is the Dirac Delta distribution. I find from Wikipedia that it equals to $\sum_{f(x)=x}\dfrac{g(x)}{|det(I-Df(x))|}$.

(ii) I also find an equation saying, $det(I-Df(x))=1-\dfrac{det A}{\lambda^2}$, where $\lambda$ is the top eigenvalue, x the top eigenvector. I can only prove it in $2\times2$ case when it is easy to define such a derivative.

Answer 1

It is true that addition is not well defined in projective space. You need to look into smooth manifolds. These are spaces (such as projective space) where it makes sense of looking at derivatives.

Answer 2

What you need in order to understand this is some differential topology. There's a lot of good books around, one of my favorite being Milnor's "topology from the differentiable viewpoint", and I have other favorites too.

Here is a brief account which might help, but to really make sense of it a good book as above will be needed.

Addition is defined in any coordinate system, and that lets you compute derivatives whenever coordinate systems have been chosen. Suppose you have a map of manifold $f:M \to N$. A choice of coordinate system in the domain $M$ and in the range $N$, as in your question, gives a local formula for the function $f$, such as the formula $(x',y')=f(x,y) = (3x/4,y/2)$ as in your example. Using that formula, you can compute the derivative as a linear transformation $D_p f$ associated to each point $p \in M$. In your question with coordinates as you have specified, $D_p f$ is given by a $2 \times 2$ matrix independent of $p$, namely $D_p f = (3/4,0;0,1/2)$. Things get interesting when you change coordinates: one can use change of coordinate theorems from multivariable calculus to relate the matrix for $D_p f$ with respect to one choice of domain and range coordinate systems to the matrix with respect to a different choice of domain and range coordinate systems.

When this is all abstracted, one gets a very nice theory of the derivative of $f:M \to N$ as a map $Df$ from the tangent space $TM$ of the domain manifold to the tangent space $TN$ of the range manifold.