Does this nonlinear feature in positive definite matrices have a name?

Anyone,

Let A be a known nxn symmetric and positive semi-definite matrix and
let x=[x1,x2,...,xn]' be a nx1 vector to be determined.

For a known positive integer N, I need to find x such that:

A*x = N*inv(x)

where I by inv(x) mean the vector where each element has been inverted,
i.e. inv(x)=[1/x1, 1/x2, ... , 1/xn].

Within the field of mathematics, does the unique vector x (given A and
N) have a name? What about properties? Can x be solved any other way
than numerically?

(This set of equations arised when setting the gradient of a particular
log-likelihood function to zero.)

Feedback appreciated.

Thanks,
Trond Jorgensen

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