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Hi. I want to normalize a discretized function ##p_{k,k'}##, to satisfy simultaneously two conditions. The normalized function ##p^*_{k,k'}## has to satisfy simultaneously:

1) ##\sum_{k=1}^{M} p^*_{k,k'} w_k=1##, for all ##k'=1,2,...,M##;

2) ##\sum_{k=1}^{M} p^*_{k,k'} w_k \hat \Omega_k \cdot \hat \Omega_{k'}=g##, for all ##k'=1,2,...,M##

The ##w_k## are weights, the ##\Omega## are versors, and I have to modify ##p_{k,k'}##, which is a given function in ##\Omega_k## in order to satisfy this both conditions.

For example, if I only had to satisfy condition 1), I would have the trivial solution:

##p^*_{k,k'}=p_{k,k'}/\sum_{k=1}^{M} p_{k',k} w_k##.

However, I'm not sure on how to proceed now that I have to satisfy both conditions simultaneously.

I was thinking that I could pose this problem this way. I could try to find the vector ##p^*_{k,k'}## considering the system of equations in matrix form. So, for example, for the first condition I would search the ##p^*_{k,k'}## that satisfy:

##(\sum_{k=1}^{M} p_{k,k'} w_k) p^*_{k,k'}=p_{k,k'}## for each ##k'##

I am trying to write these conditions in matrix form. Something like (this is clearly wrong):

##\begin{bmatrix}

p_{1,1}w_1 & p_{2,1}w_2 & p_{3,1}w_3 & \dots & p_{M,1}w_M \\

p_{1,2}w_1 & p_{2,2}w_2 & p_{3,2}w_3 & \dots & p_{M,2}w_M \\

\vdots & \vdots & \vdots & \ddots & \vdots \\

p_{1,M}w_1 & p_{2,M}w_2 & p_{3,M}w_3 & \dots & p_{M,M} w_M\\

\vdots & \vdots & \vdots & \ddots & \vdots

\end{bmatrix}

\begin{bmatrix}

p^*_{1,1} \\

p^*_{1,2} \\

\vdots\\

p^*_{1,M} \\

\vdots\

\end{bmatrix}=

\begin{bmatrix}

p_{1,1} \\

p_{1,2} \\

\vdots\\

p_{1,M} \\

\vdots\

\end{bmatrix}

##

This matrix should extend, and also include the systems for condition 2). However, I am having some trouble on figuring out the matrix coefficients and how to properly set the systems of equations.

1) ##\sum_{k=1}^{M} p^*_{k,k'} w_k=1##, for all ##k'=1,2,...,M##;

2) ##\sum_{k=1}^{M} p^*_{k,k'} w_k \hat \Omega_k \cdot \hat \Omega_{k'}=g##, for all ##k'=1,2,...,M##

The ##w_k## are weights, the ##\Omega## are versors, and I have to modify ##p_{k,k'}##, which is a given function in ##\Omega_k## in order to satisfy this both conditions.

For example, if I only had to satisfy condition 1), I would have the trivial solution:

##p^*_{k,k'}=p_{k,k'}/\sum_{k=1}^{M} p_{k',k} w_k##.

However, I'm not sure on how to proceed now that I have to satisfy both conditions simultaneously.

I was thinking that I could pose this problem this way. I could try to find the vector ##p^*_{k,k'}## considering the system of equations in matrix form. So, for example, for the first condition I would search the ##p^*_{k,k'}## that satisfy:

##(\sum_{k=1}^{M} p_{k,k'} w_k) p^*_{k,k'}=p_{k,k'}## for each ##k'##

I am trying to write these conditions in matrix form. Something like (this is clearly wrong):

##\begin{bmatrix}

p_{1,1}w_1 & p_{2,1}w_2 & p_{3,1}w_3 & \dots & p_{M,1}w_M \\

p_{1,2}w_1 & p_{2,2}w_2 & p_{3,2}w_3 & \dots & p_{M,2}w_M \\

\vdots & \vdots & \vdots & \ddots & \vdots \\

p_{1,M}w_1 & p_{2,M}w_2 & p_{3,M}w_3 & \dots & p_{M,M} w_M\\

\vdots & \vdots & \vdots & \ddots & \vdots

\end{bmatrix}

\begin{bmatrix}

p^*_{1,1} \\

p^*_{1,2} \\

\vdots\\

p^*_{1,M} \\

\vdots\

\end{bmatrix}=

\begin{bmatrix}

p_{1,1} \\

p_{1,2} \\

\vdots\\

p_{1,M} \\

\vdots\

\end{bmatrix}

##

This matrix should extend, and also include the systems for condition 2). However, I am having some trouble on figuring out the matrix coefficients and how to properly set the systems of equations.

Last edited: