Wednesday, May 19, 2021

[zxyrjjkf] similarity of vectors

how similar are vectors x and y?  dot product, cosine, project x onto y or vice versa.

how similar are vectors x and y under conditions A?  transpose(x)*A*y where A is a positive definite matrix, inner product.  if A is identity matrix, then it simplifies to dot product.

unspecified: how to encode "conditions" into the positive definite matrix A.

motivation is for a hypothetical chemistry game: how acceptable is it to substitute reagent x for reagent y under conditions A?  the conditions include the reaction being attempted.

if only positive semidefinite, there will exist nonzero vectors whose similarly to themselves is zero, i.e., as dissimilar as vectors can be.  this seems undesirable.

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