Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products

We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) $\mathrm{♯}$ and semi-tensor products $⋉$. Indeed, for each real number $t$ and two positive definite matrices $A$ and $B$ of arbitrary sizes, the $t$-weighted SGM $A{\mathrm{♢}}_{t}B$ of $A$ and $B$ is a unique positive solution $X$ of the equation

$$A^{-1}\mathrm{♯}X=(A^{-1}\mathrm{♯}B{)}^t.$$

We then established fundamental properties of the weighted SGMs based on MGMs. In addition, $(A{\mathrm{♢}}_{1/2}B{)}^2$ is positively similar to $A⋉B$ and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.