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Minimum Singular Value of a Complex Matrix

The System
Consider $$ A \in \mathbb{C}^{n \times m} $$ as well as $$ \gamma $$. A minimum singular value of a matrix $$ A $$ is greater than $$ \gamma $$ if and only if $$ AA^H > \gamma^2I $$ or $$ A^H A > \gamma^2I $$, where $$ A^H $$ is the conjugate transpose or Hermitian transpose of the matrix $$ A $$. the inequality used depends on the size of matrix $$ A $$.

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

The LMI: Minimum Singular Value of a Complex Matrix
The following LMIs can be constructed depending on the size of $$ A $$:

if $$ A \in \mathbb{C}^{n \times m} $$, where $$ n \leq m $$, then:

\begin{align} \bar{\sigma}(A) > \gamma\\ A A^{H} > \gamma^2 I \end{align}$$

Else if $$ n \geq m $$, then:

\begin{align} \bar{\sigma}(A) > \gamma\\ A^{H} A > \gamma^2 I \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}$$

This answer can also be proven using the following solution. Note that this solution only works if the matrix $$ A $$ is a square, invertible matrix: $$ \sigma_{min} = 1/||A^{-1}||_2 $$.

Implementation

 * Minimum Singular Value Example

Related LMIs

 * Maximum singular value of a complex matrix