LMIs in Control/Matrix and LMI Properties and Tools/Concatenation Of Matrices

Introduction
Matrix concatenation is the process of joining one or more matrices to make a new matrix. This process is similar for concatenating the LMIs as well.

Concatenation of LMIs
A useful property of LMIs is that multiple LMIs can be concatenated together to form a single LMI. For example, satisfying the LMIs $$ A < 0$$ and, $$B < 0$$ is equivalent to satisfying the concatenated LMI:

$$ \begin{bmatrix} A && 0 \\ 0 && B \end{bmatrix}$$ $$\begin{align} <0 \end{align}$$,

More generally, satisfying the LMIs $$Ai < 0$$, where $$i = 1,. . ., n $$ is equivalent to satisfying the concatenated LMI:

$$ \begin{bmatrix} A && 0 && 0 &&. && . && 0 \\ 0 && B && 0 &&. && . && 0 \\ 0 && 0 && C &&. && . && 0 \\ . && . && . && . && . && 0\\ . && . && . && . && . && 0 \\ 0 && 0 && 0 && 0 && 0 && A_n\\ \end{bmatrix}$$ $$\begin{align} <0 \end{align}$$,