# 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

${\displaystyle A<0}$ and,
${\displaystyle B<0}$


is equivalent to satisfying the concatenated LMI:

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

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

${\displaystyle {\begin{bmatrix}A&&0&&0&&.&&.&&0\\0&&B&&0&&.&&.&&0\\0&&0&&C&&.&&.&&0\\.&&.&&.&&.&&.&&0\\.&&.&&.&&.&&.&&0\\0&&0&&0&&0&&0&&A_{n}\\\end{bmatrix}}}$ {\displaystyle {\begin{aligned}<0\end{aligned}}},