Matrix fractional function

Question

We know that the following function is convex $$f(\mathbf{x,Y})=\mathbf{x^TY^{-1}x}$$ where $\mathbf{Y}$ is a positive definite matrix of size $n$ and $\mathbf{x}$ is a column vector of size $n$. How to use this result and formally prove that $$g(\mathbf{w})=\mathbf{b^TWA(A^TWA)^{-1}A^TWb}$$ is convex where $\mathbf{w}$ is a column vector of size $n$ and $\mathbf{b}$ is a column vector of size $n$, $\mathbf{A}$ is a matrix of size $n\times m$ and $\mathbf{W}=diag(\mathbf{w})$. Thanks in advance.

Answer 1

Convexity is closed under linear transformations, and you have $f(A^TWb, A^TWA)$.