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list manipulation - Replace elements of a matrix with zeros using another matrix


I have a matrix Mat1 whos columns are ordered as :


Col = Flatten[Outer[{#2, #1} &, {0, 1}, Delete[Range[-2, 2, 1], 3]], 1]

and the rows are ordered as:


Rows = Flatten[Outer[{#1, #2} &, Delete[Range[-1, 1, 1], 2], Delete[Range[-1, 1, 1], 2]], 1]

The matrix is given as:


Mat1 = Outer[f[Flatten[{#1, #2}]] &, Col, Rows, 1]



$$ \left( \begin{array}{cccc} f(\{-2,0,-1,-1\}) & f(\{-2,0,-1,1\}) & f(\{-2,0,1,-1\}) & f(\{-2,0,1,1\}) \\f(\{-1,0,-1,-1\}) & f(\{-1,0,-1,1\}) & f(\{-1,0,1,-1\}) & f(\{-1,0,1,1\}) \\f(\{1,0,-1,-1\}) & f(\{1,0,-1,1\}) & f(\{1,0,1,-1\}) & f(\{1,0,1,1\}) \\f(\{2,0,-1,-1\}) & f(\{2,0,-1,1\}) & f(\{2,0,1,-1\}) & f(\{2,0,1,1\}) \\ f(\{-2,1,-1,-1\}) & f(\{-2,1,-1,1\}) & f(\{-2,1,1,-1\}) & f(\{-2,1,1,1\}) \\ f(\{-1,1,-1,-1\}) & f(\{-1,1,-1,1\}) & f(\{-1,1,1,-1\}) & f(\{-1,1,1,1\}) \\ f(\{1,1,-1,-1\}) & f(\{1,1,-1,1\}) & f(\{1,1,1,-1\}) & f(\{1,1,1,1\}) \\ f(\{2,1,-1,-1\}) & f(\{2,1,-1,1\}) & f(\{2,1,1,-1\}) & f(\{2,1,1,1\}) \end{array} \right)$$



Another matrix Mat2 with a fewer elements is given as:


Mat2 = Outer[f[Flatten[{#1 , #2}]] &, {{-2, 0}, {1, 0}, {2, 1}}, {{-1, 1}, {1, -1}, {1, 1}}, 1]


$$\left( \begin{array}{ccc} f(\{-2,0,-1,1\}) & f(\{-2,0,1,-1\}) & f(\{-2,0,1,1\}) \\ f(\{1,0,-1,1\}) & f(\{1,0,1,-1\}) & f(\{1,0,1,1\}) \\ f(\{2,1,-1,1\}) & f(\{2,1,1,-1\}) & f(\{2,1,1,1\}) \end{array} \right)$$




I want to make all the elements of the first matrix (Mat1) equal to zero that are not same as elements of Mat2. This will create a kind of a sparse matrix like:



$$\left( \begin{array}{cccc} 0 & f(\{-2,0,-1,1\}) & f(\{-2,0,1,-1\}) & f(\{-2,0,1,1\}) \\ 0 & 0 & 0 & 0 \\ 0 & f(\{1,0,-1,1\}) & f(\{1,0,1,-1\}) & f(\{1,0,1,1\}) \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & f(\{2,1,-1,1\}) & f(\{2,1,1,-1\}) & f(\{2,1,1,1\}) \end{array} \right)$$



How can I do this? (and for any general Range of col and Rows)


Edit: Building Mat1 with Outer for a large list of Rows and Col causes my system to hang and removes all variables from Mathematica. Outer carries out the arrangement of the elements at the positions I need in the matrix but it doesn't work for very big dimensions. I am only interested in the final sparse matrix with the elements at the positions dictated by the list of Rows and Col. Is there any way to get the final matrix by somehow using Rows and Col without the need to build Mat1?.



Answer



Update: An approach to use Col, Rows and the indices used to create Mat2 to get the desired sparse array without creating Mat1:


c2 = {{-2, 0}, {1, 0}, {2, 1}};
r2 = {{-1, 1}, {1, -1}, {1, 1}};

positions = Tuples[{Flatten@Position[Col, #]& /@ c2, Flatten@Position[Rows, #]& /@ r2}];

SparseArray[positions -> Flatten @ Mat2, Length /@ {Col, Rows}] // MatrixForm // TeXForm


$\left( \begin{array}{cccc} 0 & f(\{-2,0,-1,1\}) & f(\{-2,0,1,-1\}) & f(\{-2,0,1,1\}) \\ 0 & 0 & 0 & 0 \\ 0 & f(\{1,0,-1,1\}) & f(\{1,0,1,-1\}) & f(\{1,0,1,1\}) \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & f(\{2,1,-1,1\}) & f(\{2,1,1,-1\}) & f(\{2,1,1,1\}) \\ \end{array} \right)$



If you like, you can use f directly without having to use Mat2:


SparseArray[positions -> (f[Flatten[{Col[[#[[1]]]], Rows[[#[[2]]]]}]] & /@ 
     positions), Length /@ {Col, Rows}] // MatrixForm // TeXForm



same result



Original answer:


SparseArray


You can define a function f0 that takes two matrices as input and returns a SparseArray with the desired entries:


ClearAll[f0]
f0[m1_, m2_] := Module[{pos = Position[m1, Alternatives @@ Flatten[m2]]},
   SparseArray[pos -> Extract[m1, pos], Dimensions[m1]]];


f0[Mat1, Mat2] // MatrixForm // TeXForm


$\left( \begin{array}{cccc} 0 & f(\{-2,0,-1,1\}) & f(\{-2,0,1,-1\}) & f(\{-2,0,1,1\}) \\ 0 & 0 & 0 & 0 \\ 0 & f(\{1,0,-1,1\}) & f(\{1,0,1,-1\}) & f(\{1,0,1,1\}) \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & f(\{2,1,-1,1\}) & f(\{2,1,1,-1\}) & f(\{2,1,1,1\}) \\ \end{array} \right)$



ReplaceAll


Alternatively, define a function f1 that gives 0 for any input except one that matches elements of Mat2 and use it with ReplaceAll:


ClearAll[f1]
f1[x : Alternatives @@ Flatten[Mat2]] := x

f1[_] := 0;

Mat1 /. a_f :> f1[a] // MatrixForm // TeXForm


same result



You can also Map f1 on Mat1 at level 2:


Map[f1, Mat1, {2}] // MatrixForm // TeXForm



same result



or use it to construct the result directly without creating Mat:


Outer[f1@f[Flatten[{#1, #2}]] &, Col, Rows, 1] // MatrixForm // TeXForm


same result



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