Bijection of transformations

Question

Is there a bijection from a row vector x = [1xn] to a symmetric positive definite (SPD) matrix [n x n]?

Here is what I have:

1. C2 = x^T.x gives a n by n square matrix
2. C2 = 0.5(C2 + C2^T) [to make it symmetric]
3. C2 = C2 + epsilon * np.identity (C2.shape[0]) [ eigenvalues are all zero and positive so by adding this negligible epsilon, all eigenvalues are now positive ]

However, when I trace back, x^T.x will not be able to give a bijection. Are there any other methods?