LMIs in Control/Matrix and LMI Properties and Tools/Singular Value Bounds of a Matrix through LMIs

The singular values of a real-valued matrix can be bounded through the use of LMI techniques.

Required Data
The real-valued matrix of interest, $$ A \in \mathbb{R}^{n \times m} $$, is needed.

Maximum Singular Value
Consider matrix $$ A \in \mathbb{R}^{n \times m} $$ and $$ \gamma \in \mathbb{R} $$ such that $$ \gamma > 0 $$.

The maximum singular value of $$ A $$ is strictly less than $$ \gamma $$ (that is, $$ \bar{\sigma}(A) < \gamma $$ ) if and only if $$ AA^T < \gamma^2 I $$.

Using the Schur complement, we see that $$ AA^T < \gamma^2 I $$ is equivalent to

$$ \begin{align} \begin{bmatrix} \ \gamma I & A \\ \ A^T & \gamma I \\ \end{bmatrix} \ > 0, \end{align} $$

Which is also equivalent to: $$ \bar{\sigma}(A)<\gamma $$ if and only if $$ A^TA < \gamma^2 I $$ or

$$ \begin{align} \begin{bmatrix} \ \gamma I & A^T \\ \ A & \gamma I \\ \end{bmatrix} \ > 0 . \end{align} $$

Minimum Singular Value
Consider matrix $$ A \in \mathbb{R}^{n \times m} $$ and $$ \nu \in \mathbb{R} $$ such that $$ \nu \geq 0 $$.

If $$n \leq m$$, then the minimum singular vale of $$ A $$ is strictly greater than $$ \nu $$ (that is, The maximum singular value of $$ A $$ is strictly less than $$ \gamma $$ (that is, $$ \underline{\sigma}(A) > \nu $$ ) if and only if $$AA^T < \nu^2 I $$. If $$ m \leq n $$, then $$ \underline{\sigma}(A) > \nu $$ if and only if $$A^TA > \nu^2 I$$.

Conclusion:
LMI techniques can be used to establish bounds on the the singular values of any real-valued matrix.

Related LMIs
Singular Value Bounds for complex-valued matrices

https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Singular_Value_of_a_Complex_Matrix

https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Singular_Value_of_a_Complex_Matrix