Let f: \R^n -> \R^n be a function. What would you call the function g: x -> x' f(x) ? Furthermore, if F:\R^n -> \R^n is a linear operator and D \in \R^{n X m} a matrix, how would you refer to the operator G: D -> D' F (D) ? Does this appear in a particular physics or engineering context? Note, I use dash ' to mean transpose.
Hi Peter, thanks, I've updated the title with "inner-product", also interesting matrix representation of a tensor.
Some further clarification, in my work, I'm interested in building operators with a desired action, such as finding a linear operator T: \R^n -> \R^n with the same action as F over a subspace, namely T: D -> F(D), among other desired properties. In determining T, the operator G: D -> D' F(D) crops up all the time.
Just to add to Peter's first answer: It's technically not even a quadratic form if f is nonlinear. I don't know off-hand of any specific name for it though other than, as Peter said, it's .
This is a an optimization problem, at this stationary point we obtain maximum visibility or decision boundary points or roots. It is very fundamental concept to pattern analysis, in particular classification problem.
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