Can you please help me solve the following nonlinear equation to determine the value of the vector $z$ :
$$
\boldsymbol{a}=\boldsymbol{z}^{2} \odot \boldsymbol{K}*\boldsymbol{z}^{-1}$$
Where:
- $\boldsymbol{a}$ is a given vector with all positive entries.
- $K$ is an element-wise positive, symmetric, and positive semi-definite matrix.
- $\odot$ represents the element-wise product.
- the exponents represent elementwise exponentiation
- $*$ represents matrix multiplication.

Is there another approach or method to solve this equation?

The equation above can be written as follows for each element of vectors $\boldsymbol{a}$, $\boldsymbol{z}$, and matrix $\boldsymbol{K}$
$a_i=\sum_j z_i^2 K_{i j} z_j^{-1}$