- #1
brydustin
- 205
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Given a set of vectors {v_ j } = {v_1, ... v_N} and I wish to transform each vector in the following manner
v_ i ' = Sum_k=1 to N ( c_ k,i) * v_i where c_ k,i is a scalar and what we are trying to solve for.
such that the sum of the distances squared between each pair of transformed vectors is maximized.
Then we would like to solve dD/dc = 0 where D is the sum of the distances squared and c are the scalars in matrix form. I'm not sure how to think of the c's except I could define a diagonal matrix whose entries are the sum of the respective scalars (i.e. i-th row is Sum_k=1 to N ( c_ k,i)) and multiply it by a vector (tensor) whose elements are the v_ j's so that the new vector has as its elements the transformed vectors.
v_ i ' = Sum_k=1 to N ( c_ k,i) * v_i where c_ k,i is a scalar and what we are trying to solve for.
such that the sum of the distances squared between each pair of transformed vectors is maximized.
Then we would like to solve dD/dc = 0 where D is the sum of the distances squared and c are the scalars in matrix form. I'm not sure how to think of the c's except I could define a diagonal matrix whose entries are the sum of the respective scalars (i.e. i-th row is Sum_k=1 to N ( c_ k,i)) and multiply it by a vector (tensor) whose elements are the v_ j's so that the new vector has as its elements the transformed vectors.
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