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 $\boldsymbol{K}$ $a_i=\sum_j z_i^2 K_{i j} z_j^{-1}$