I was going through Boyd & Vandenberghe's Convex Optimization. On page 38, the authors mentioned that the solution set of a linear matrix inequality (LMI) is convex.
$$ A(x) := x_1 A_1 + \dots + x_n A_n \preceq B $$
where $A_1, \dots, A_n, B \in \mathbb{S}^m$, is called an LMI in $x$. They also gave a brief explanation where they mentioned that this is because
it is the inverse image of the positive semi-definite cone under the affine function.
I could not figure out what would be the affine function that they mentioned.
Best Answer
The affine function is $T(x) = B - x_1 A_1 - \cdots - x_n A_n $.
The solution set to your LMI can be described as \begin{equation} \{ x \mid T(x) \succeq 0 \} = T^{-1}(S^m_+), \end{equation} where $S^m_+$ is the positive semidefinite cone in $\mathbb R^{m\times m}$.
Further details:
If we view $A_1,\ldots,A_n$ and $B$ as column vectors in $\mathbb R^{m^2}$, then \begin{equation} T(x) = \underset{\substack{\Bigg \uparrow \\m^2 \times 1}}{B} - \underset{\substack{\Bigg \uparrow \\ m^2 \times n}}{A} \underset{\substack{\uparrow \\ n \times 1}}{x} \end{equation} where \begin{equation} A = \begin{bmatrix} A_1 & A_2 & \cdots & A_n \end{bmatrix}. \end{equation} In this equation, the $A$ is multiplied by $x$ using ordinary matrix multiplication.