LMIs in Control/pages/Maximum Singular Value of a Complex Matrix

Maximum Singular Value of a Complex Matrix

The System
Consider $$ A \in \mathbb{R}^{n \times m} $$ as well as $$ \gamma $$. A maximum singular value of a matrix $$ A $$ is less than $$ \gamma $$ if and only if $$ AA^H < \gamma^2I $$, where $$ A^H $$ is the conjugate transpose or Hermitian transpose of the matrix $$ A $$.

The Data
The matrix $$ A $$ is the only data required.

The LMI: Maximum Singular Value of a Complex Matrix
Using the Shur complement procedure, the following LMIs can be constructed:



\begin{align} \bar{\sigma}(A) < \gamma\\ \begin{bmatrix} \gamma I & A \\ A^H & \gamma I\end{bmatrix}&>0\\ \end{align}$$

The following LMI is also equivalent:



\begin{align} \bar{\sigma}(A) < \gamma\\ \begin{bmatrix} \gamma I & A^H \\ A & \gamma I\end{bmatrix}&>0\\ \end{align}$$

Conclusion:
The results from this LMI will give the maximum complex value of the matrix $$ A $$:

\begin{align} \bar{\sigma}(A) < \gamma \end{align}$$

Implementation

 * Maximum Singular Value Example

Related LMIs

 * Minimum singular value of a complex matrix