Tensors are square symmetric positive-definite matrix. They are surprisingly in 1-to-1 mapping with symmetric matrices through matrix exponentiation (exp.log computed on the diagonal elements of the spectral decomposition). I said surprisingly because tensors are symmetric and therefore a proper subset of the set they are in bijection with. Yes, nobody knows what is going on at infinity, and Cantor set theory and life was not so happy due to this… Anyway, obviously symmetric matrices are closed by addition and scalar multiplication, so that it is a vector space. Now using the bjiection tensor-symmetric matrix, we deduce a vector space for tensors. Euclidean distance (or other norm induced distance) hold, and we can transpose that to distances on tensors… These are called log-euclidean metrics, for distance metrics on matrix logarithm. And yes the metric with triangle inequality is derived from Cauchy-Schwarz inner product ineq.
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