Homogeneous Inner Product $\langle \cdot,\cdot\rangle_{\mathbf{ d } }$
Let a linear continuous dilation $\mathbf{d}$ in $\mathbb{R^n}$ be strictly monotone with respect to a weighted Euclidean norm $\|x\|=\sqrt{x^{\top}Px}$. The homogeneous inner product is defined as follows $$ \langle x,y\rangle_{\mathbf{d}}=\Phi^{\top}(x)P\Phi(y) $$ where $x,y\in \mathbb{R}^n$ and $\Phi$ is a homogeneous homeomorphism on $\mathbb{R}^{n}$.$$ \quad $$
Computation of the homogeneous inner product
The function ${\color{red} { \texttt{hinner }} } $ computes the homogeneous inner product $\langle x,y\rangle_{\mathbf{d}}$ of the vectors $x$ and $y$.
- $\textbf{Input parameters}: {\color{blue}x,y} \text{ (vectors)}, {\color{blue}{G_{\mathbf{ d } }}} \text{ (generator of linear dilation)}, {\color{blue}P}\text{ (shape matrix of weighted Euclidean norm)} $
- $ \textbf{Output parameters}: {\color{magenta}{\langle x,y\rangle_{\mathbf{d}}}} $
${\color{red} { \texttt{demo_halgebra.m}} } $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Example of demonstrating operators in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tol=1e-6; % compuational tolerance %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Shape matrix of the wighted Euclidean norm %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P=[1 -1; -1 2]; if norm(P-P')>tol disp('Error: shape matrix P must be symmetric'); return; end; if min(real(eig(P)))< tol disp('Error: shape matrix P must be positive definite'); return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Generator of the linear dilation in R^2 %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gd=[3 0; 1 1]; %monotonicy check if min(real(eig(P*Gd+Gd'*P)))< tol disp('Error: dilation must be monotone'); return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Sum of vectors in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['--Addition in homogeneous Euclidean space R^2_d--']); % computed by bisection method x=[1;-1]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); y=[-0.5;1]; % vector y in R^2 disp(['Vector y = [', num2str(y(1)),';',num2str(y(2)),']']); z=x+y; disp(['x+y in R^2 =[', num2str(z(1)),';',num2str(z(2)),']']); % sum of vectors in R^2 % computed by bisection method z=hadd(x,y,Gd,P); disp(['x+y in R^2_d =[', num2str(z(1)),';',num2str(z(2)),']']); % sum of vectors in homogeneous Euclidean space R^2_d % computed by bisection method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Multiplication of a vector by a scalar in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['--Multiplication by a scalar in homogeneous Euclidean space R^2_d--']); % computed by bisection method a=0.7; disp(['Scalar a= ', num2str(a)]); x=[1;-5]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); z=a*x; disp(['a*x in R^2 =[', num2str(z(1)),';',num2str(z(2)),']']); % multiplication by a scalar in R^2 % computed by bisection method z=hdot(a,x,Gd); disp(['a*x in R^2_d =[', num2str(z(1)),';',num2str(z(2)),']']); % multiplication by a scalar in homogeneous Euclidean space R^2_d % computed by bisection method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Inner product in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P=eye(2); % for comparison with canonical inner product in R^2 disp(['--Inner product in homogeneous Euclidean space R^2_d--']); % computed by bisection method x=[2;0]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); y=[0;1]; % vector y in R^2 disp(['Vector y = [', num2str(y(1)),';',num2str(y(2)),']']); p=x'*P*y; disp(['< x,y > in R^2 =', num2str(p)]); % inner product of vectors in R^2 % computed by bisection method p=hinner(x,y,Gd,P); disp(['< x,y > in R^2_d =', num2str(p)]); % inner product of vectors in homogenous Euclidean space R^2_d % computed by bisection method