It is well known that the matrix fractional function $f(\mathbf{w},\boldsymbol{\Omega})=\mathbf{w}^T\boldsymbol{\Omega}^{-1}\mathbf{w}$ is jointly convex with respect to $\mathbf{w}$ and $\boldsymbol{\Omega}$ for $\mathbf{w}\in\mathbb{R}^n$ and $\boldsymbol{\Omega}\in\mathbb{S}_+^{n\times n}$, where $\mathbb{S}_+^{n\times n}$ denotes the set of all $n\times n$ positive definite matrix.
Now I want to prove it based on the definition of convex functions. That is, I need to prove that the following inequality holds for any $\mathbf{w}_1,\mathbf{w}_2\in\mathbb{R}^n$, $\boldsymbol{\Omega}_1,\boldsymbol{\Omega}_2\in\mathbb{S}^{n\times n}_+$, and $\alpha\in[0,1]$: $$\alpha\mathbf{w}_1^T\boldsymbol{\Omega}_1^{-1}\mathbf{w}_1+\beta\mathbf{w}_2^T\boldsymbol{\Omega}_2^{-1}\mathbf{w}_2-(\alpha\mathbf{w}_1+\beta\mathbf{w}_2)^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}(\alpha\mathbf{w}_1+\beta\mathbf{w}_2)\ge 0,$$ where $\beta=1-\alpha$. In order to prove this inequality, I first simplify the left-hand side as \begin{align*} &\alpha\mathbf{w}_1^T\boldsymbol{\Omega}_1^{-1}\mathbf{w}_1+\beta\mathbf{w}_2^T\boldsymbol{\Omega}_2^{-1}\mathbf{w}_2-(\alpha\mathbf{w}_1+\beta\mathbf{w}_2)^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}(\alpha\mathbf{w}_1+\beta\mathbf{w}_2)\\ =&(\alpha\mathbf{w}_1^T\boldsymbol{\Omega}_1^{-1}\mathbf{w}_1-\alpha^2\mathbf{w}_1^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_1)+(\beta\mathbf{w}_2^T\boldsymbol{\Omega}_2^{-1}\mathbf{w}_2-\beta^2\mathbf{w}_2^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_2)\\ &-2\alpha\beta\mathbf{w}_1^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_2\\ =&\alpha\beta\mathbf{w}_1^T\boldsymbol{\Omega}_1^{-1}\boldsymbol{\Omega}_2(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_1+\alpha\beta\mathbf{w}_2^T\boldsymbol{\Omega}_2^{-1}\boldsymbol{\Omega}_1(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_2\\ &-2\alpha\beta\mathbf{w}_1^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_2\\ =&\alpha\beta\mathbf{w}_1^T(\alpha\boldsymbol{\Omega}_1\boldsymbol{\Omega}_2^{-1}\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_1)^{-1}\mathbf{w}_1+\alpha\beta\mathbf{w}_2^T(\alpha\boldsymbol{\Omega}_2+\beta\boldsymbol{\Omega}_2\boldsymbol{\Omega}_1^{-1}\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_2\\ &-2\alpha\beta\mathbf{w}_1^T(\alpha\boldsymbol{\Omega}_1+\beta\boldsymbol{\Omega}_2)^{-1}\mathbf{w}_2 \end{align*} I cannot prove the inequality based on this simplification. Is there any way to prove the inequality? Thanks.