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I'm trying to expand a function of a bilinear map in a power series [itex]f({\bf x^{T}Ax})[/itex]. It isn't quite a matrix function because it takes a matrix and a vector and maps them into a scalar. I'd like to expand it into powers of [itex]\bf A[/itex], but still preserve the function as a scalar. As far as I can tell, a matrix power series takes a matrix as an input (like this one) but outputs a matrix (unlike this problem). I'm not sure how to proceed.

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