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

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


Minimum Singular Value of a Complex Matrix


The SystemEdit

Consider   as well as  . A minimum singular value of a matrix   is greater than   if and only if   or  , where   is the conjugate transpose or Hermitian transpose of the matrix  . the inequality used depends on the size of matrix  .

The DataEdit

The matrix   is the only data required.

The LMI: Minimum Singular Value of a Complex MatrixEdit

The following LMIs can be constructed depending on the size of  :

if  , where  , then:

 

Else if  , then:

 

Conclusion:Edit

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

 

This answer can also be proven using the following solution. Note that this solution only works if the matrix   is a square, invertible matrix:  .


ImplementationEdit

% Minimum Singular Value of Complex Matrix
% -- EXAMPLE --

%Clears all variables
clear; clc; close all;

%SDPVAR variables
gam = sdpvar(1);

%Example Matrix A
A = rand(6,6)+rand(6,6)*1i;

%Constraints
Fc = ( A'*A >= gam*eye(6));

%Objective function
obj=-gam;

%options
opt = sdpsettings('solver','sedumi');

%Optimization
optimize(Fc,obj,opt)

%Displays output
fprintf('\nValue of Min singular value: ')
disp(value(sqrt(gam)))

fprintf('\nMATLAB verified output: ')
disp(1/norm(norm(A^(-1))))

Related LMIsEdit

External LinksEdit

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