How to find a vector x where X=x*x' is a known square matrix of order n?

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Avik Pradhan

If x=(x_1,x_2,...,x_n), then diagonal entries of the matrix X_ii=(x_i)^2. Hence we can find the vector componant x_i=sqrt(X_ii).

Guilherme Martignago Zilli

Is is only possible when X is rank one.

Otherwise, being r=rank(X), X will be a sum of r products x(1:r)*x(1:r)T.

Also, what Avik Pradhan said may not be very consistent.

f(x) = x^2 is not bijective, thus analyzing only the squares of the diagonal elements can lead to a wrong result.