Gershgorin's Circle: how can I find the intersection of the union of circles?

Good Morning, I'm trying to do a program that calculates the Gerschgorin's circles (for rows and for columns).

The following program creates the Gerschgorin's Circle for matrix A and plot them (in the first figure) and for the matrix A^T (in the second figure).

What I want to do is to find the union of the circles of the matrix A, find the union of the circles of the matrix A^T and plot the intesection between the two unions. I tried with the polyshape and the polyout but it seems not to work.

The program it's the following:

% gerschgorin.m

function gerschgorin(A)

if size(A,1) ~= size(A,2)

error('La matrice deve essere quadrata.'); %The matrix must be square

return;

end

%FIGURE 1:I calculate and represent the circles associated with matrix A;

figure;

for i=1:size(A,1)

% The circle's center is (h,k) where h is the real part of A(i,i) and k is the imaginary part of A(i,i);

h=real(A(i,i)); k=imag(A(i,i));

% I find the radius of the circle;

r=0;

for j=1:size(A,1)

if i ~= j

r=r+(norm(A(i,j)));

end

end

t=0:0.01:2*pi;

plot( r*cos(t)+h, r*sin(t)+k ,'-'); %I use the polar coordinates

hold on

c=plot( h, k,'bo');

title('Circles of the matrix A')

end

% Now we plot the actual eigenvalues of the matrix;

ev=eig(A);

for i=1:size(ev)

rev=plot(real(ev(i)),imag(ev(i)),'ro');

end

xline(0); yline(0);

axis equal;

legend([c rev],'Centri','Autovalori')

grid on;

xlabel('Parte Reale'); ylabel('Parte Immaginaria')

%FIGURE 2:I calculate and represent the circles associated with matrix A^T;

figure;

for i=1:size(A,1)

h=real(A(i,i)); k=imag(A(i,i));

r=0;

for j=1:size(A,1)

if i ~= j

r=r+(norm(A(j,i)));

end

end

t=0:0.01:2*pi;

plot( r*cos(t)+h, r*sin(t)+k ,'-'); hold on;

c=plot( h, k,'bo');

title('Cerchi associati alla matrice \itA^{T}')

end

%Eigenvalues

ev=eig(A);

for i=1:size(ev)

rev=plot(real(ev(i)),imag(ev(i)),'ro');

end

xline(0); yline(0);

axis equal;

legend([c rev],'Centri','Autovalori')

grid on;

xlabel('Parte Reale'); ylabel('Parte Immaginaria')

% FIGURE 3: Intersection

figure;

for i =1:size (A,1)

h = real (A(i,i));k=imag(A(i,i)) ;

r1 =0; r2 =0;

for j =1: size(A,1)

if i ~=j

r1=r1+(norm(A(i,j)));

r2=r2+(norm(A(j,i)));

x1=r1*cos(t)+h;

y1=r1*sin(t)+k,'-';

x2=r2*cos(t)+h;

y2=r2*sin(t)+k,'-';

pgon1=polyshape(x1,y1);

pgon2=polyshape(x2,y2);

polyout1=union(pgon1);

polyout2=union(pgon2);

end

end

plot(intersect(polyout1,polyout2),'EdgeColor','red')

hold on;

c = plot (h,k,'bo');

%title ( ' Intersezione dei due insiemi di cerchi ')

end

%Eigenvalues

ev=eig(A);

for i=1:size(ev)

rev=plot(real(ev(i)),imag(ev(i)),'ro');

end

xline(0); yline(0);

axis equal;

grid on;

xlabel('Parte Reale'); ylabel('Parte Immaginaria')

legend([c rev],'Centri','Autovalori')

end

THANK YOU for everyone who's going to answer me!

Answer 1

[KALYAN ACHARJYA]

warning off;
% gerschgorin.m
A=magic(5);
gerschgorin(A)

function gerschgorin(A)
    if size(A,1) ~= size(A,2) 
        error('La matrice deve essere quadrata.'); % The matrix must be square.
        return;
    end

    % FIGURE 1: Calculate and represent the circles associated with matrix A.
    figure; 
    for i = 1:size(A,1) 
        % The circle's center is (h, k) where h is the real part of A(i,i) and k is the imaginary part of A(i,i).
        h = real(A(i,i));
        k = imag(A(i,i)); 

        % Find the radius of the circle.
        r = 0;
        for j = 1:size(A,1)
            if i ~= j 
                r = r + abs(A(i,j));
            end    
        end 

        t = 0:0.01:2*pi; 
        plot(r*cos(t)+h, r*sin(t)+k, '-'); % Use polar coordinates.
        hold on;
        c = plot(h, k, 'bo'); 
    end

    % Plot the actual eigenvalues of the matrix.
    ev = eig(A); 
    rev = plot(real(ev), imag(ev), 'ro');
    xline(0);
    yline(0);
    axis equal;
    legend([c rev], 'Centri', 'Autovalori');
    grid on;
    xlabel('Parte Reale');
    ylabel('Parte Immaginaria');
    title('Circles of matrix A');

    % FIGURE 2: Calculate and represent the circles associated with matrix A^T.
    figure;
    for i = 1:size(A,1) 
        h = real(A(i,i));
        k = imag(A(i,i)); 

        r = 0;
        for j = 1:size(A,1)
            if i ~= j 
                r = r + abs(A(j,i));
            end    
        end 

        t = 0:0.01:2*pi;
        plot(r*cos(t)+h, r*sin(t)+k, '-');
        hold on;
        c = plot(h, k, 'bo');
    end

    % Plot the actual eigenvalues of the matrix.
    ev = eig(A);
    rev = plot(real(ev), imag(ev), 'ro');
    xline(0);
    yline(0);
    axis equal;
    legend([c rev], 'Centri', 'Autovalori');
    grid on;
    xlabel('Parte Reale');
    ylabel('Parte Immaginaria');
    title('Circles of matrix A^T');

    % FIGURE 3: Intersection of the union of circles for A and A^T.
    figure;
    for i = 1:size(A,1)
        h = real(A(i,i));
        k = imag(A(i,i));
        r1 = 0;
        r2 = 0;
        x = [];
        y = [];

        for j = 1:size(A,1)
            if i ~= j
                r1 = r1 + abs(A(i,j));
                r2 = r2 + abs(A(j,i));
                x1 = r1*cos(t) + h;
                y1 = r1*sin(t) + k;
                x2 = r2*cos(t) + h;
                y2 = r2*sin(t) + k;
                x = [x, x1];
                y = [y, y1];
            end
        end

        polyout1 = polyshape(x, y);
        polyout2 = polyshape(x2, y2);
        intersection = intersect(polyout1, polyout2);
        plot(intersection, 'EdgeColor', 'red');
        hold on;
        c = plot(h, k, 'bo');
    end

    % Plot the actual eigenvalues of the matrix.
    ev = eig(A); 
    rev = plot(real(ev), imag(ev), 'ro');
    xline(0);
    yline(0);
    axis equal;
    grid on;
    xlabel('Parte Reale');
    ylabel('Parte Immaginaria');
    title('Intersection of the union of circles for A and A^T');
    legend([c rev], 'Centri', 'Autovalori');
end

[Federica Mina]

Thank you for your answer!! I tried the program with the following matrix but it doesn't work as it should.

-2 0 -6

2 -2 0

0 2 -2

[Federica Mina]

@KALYAN ACHARJYA sorry I tagg you so you can watch my comment. I am a beginner of this forum so I don't know how it works. thanks

[KALYAN ACHARJYA]

warning off;
% gerschgorin.m
A=[-2     0    -6
     2    -2     0
     0     2    -2];
gerschgorin(A)

function gerschgorin(A)
    if size(A,1) ~= size(A,2) 
        error('La matrice deve essere quadrata.'); % The matrix must be square.
        return;
    end

    % FIGURE 1: Calculate and represent the circles associated with matrix A.
    figure; 
    for i = 1:size(A,1) 
        % The circle's center is (h, k) where h is the real part of A(i,i) and k is the imaginary part of A(i,i).
        h = real(A(i,i));
        k = imag(A(i,i)); 

        % Find the radius of the circle.
        r = 0;
        for j = 1:size(A,1)
            if i ~= j 
                r = r + abs(A(i,j));
            end    
        end 

        t = 0:0.01:2*pi; 
        plot(r*cos(t)+h, r*sin(t)+k, '-'); % Use polar coordinates.
        hold on;
        c = plot(h, k, 'bo'); 
    end

    % Plot the actual eigenvalues of the matrix.
    ev = eig(A); 
    rev = plot(real(ev), imag(ev), 'ro');
    xline(0);
    yline(0);
    axis equal;
    legend([c rev], 'Centri', 'Autovalori');
    grid on;
    xlabel('Parte Reale');
    ylabel('Parte Immaginaria');
    title('Circles of matrix A');

    % FIGURE 2: Calculate and represent the circles associated with matrix A^T.
    figure;
    for i = 1:size(A,1) 
        h = real(A(i,i));
        k = imag(A(i,i)); 

        r = 0;
        for j = 1:size(A,1)
            if i ~= j 
                r = r + abs(A(j,i));
            end    
        end 

        t = 0:0.01:2*pi;
        plot(r*cos(t)+h, r*sin(t)+k, '-');
        hold on;
        c = plot(h, k, 'bo');
    end

    % Plot the actual eigenvalues of the matrix.
    ev = eig(A);
    rev = plot(real(ev), imag(ev), 'ro');
    xline(0);
    yline(0);
    axis equal;
    legend([c rev], 'Centri', 'Autovalori');
    grid on;
    xlabel('Parte Reale');
    ylabel('Parte Immaginaria');
    title('Circles of matrix A^T');

    % FIGURE 3: Intersection of the union of circles for A and A^T.
    figure;
    for i = 1:size(A,1)
        h = real(A(i,i));
        k = imag(A(i,i));
        r1 = 0;
        r2 = 0;
        x = [];
        y = [];

        for j = 1:size(A,1)
            if i ~= j
                r1 = r1 + abs(A(i,j));
                r2 = r2 + abs(A(j,i));
                x1 = r1*cos(t) + h;
                y1 = r1*sin(t) + k;
                x2 = r2*cos(t) + h;
                y2 = r2*sin(t) + k;
                x = [x, x1];
                y = [y, y1];
            end
        end

        polyout1 = polyshape(x, y);
        polyout2 = polyshape(x2, y2);
        intersection = intersect(polyout1, polyout2);
        plot(intersection, 'EdgeColor', 'red');
        hold on;
        c = plot(h, k, 'bo');
    end

    % Plot the actual eigenvalues of the matrix.
    ev = eig(A); 
    rev = plot(real(ev), imag(ev), 'ro');
    xline(0);
    yline(0);
    axis equal;
    grid on;
    xlabel('Parte Reale');
    ylabel('Parte Immaginaria');
    title('Intersection of the union of circles for A and A^T');
    legend([c rev], 'Centri', 'Autovalori');
end

[Federica Mina]

@KALYAN ACHARJYA Yes exactly what I meant. My purpose is to plot the intersection between the union of the circles of matrix A and the union of circles of matrix A^T. Then the intersection must to include all the eigenvalues of the matrix A. So in this case two of the eigenvalues are outside of the circle. Should I change somenthing else in the program? Thank you for your answer by the way.