MATLAB Problem. How to design matrix from eigenvalues

Question

Panos Kerezoudis 2023-1-23

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1899230-matlab-problem-how-to-design-matrix-from-eigenvalues

编辑： Matt J 2023-1-24

This is a previously posted problem that I am working on but I can't find any solutions online. Here is the description:

Write a function that takes one input argument n and outputs a (n x n) square matrix A with the following properties:

A has an eigenvalue of 3
all elements of A differ more than .5 from each other

Here is what I have tried so far. I am having difficulty with setting up the specific eigenvalue.

function A = matrix_design(n)
A=randn(n); % set up the matrix
[V,D]=eig(A) % use built-in function for eigendecomposition
abs(A-A.')>0.5 % specify elements of A to differ by 0.5
end
disp(A)

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Torsten 2023-1-23

I don't understand how the code could lead to a solution of the problem.

Walter Roberson 2023-1-23

在 MATLAB Online 中打开

Is it possible at all?

d = sym(2)/3

d =

syms c

M = [c+0*d, c+1*d, c+2*d

c+3*d, c+4*d, c+5*d

c+7*d, c+9*d, c+8*d]

M =

[V,D] = eig(M);

c0 = solve(D(1)==3, c)

c0 =

Mnum = double(subs(M, c, c0))

Mnum = 3×3

-2.6566 -1.9899 -1.3232 -0.6566 0.0101 0.6768 2.0101 3.3434 2.6768

eig(Mnum)

ans = 3×1

-2.6320 -0.3377 3.0000

... I guess so, at least for the 3 x 3 case -- and assuming that "A has an eigenvalue of 3" means that at least one of the eigenvalues of the matrix is 3

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Answer 1

Matt J 2023-1-23

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https://ww2.mathworks.cn/matlabcentral/answers/1899230-matlab-problem-how-to-design-matrix-from-eigenvalues#answer_1154805

编辑：Matt J 2023-1-24

在 MATLAB Online 中打开

mindist=@(A) min(sqrt(pdist(real(A(:))).^2+pdist(imag(A(:))).^2));
n=5;
Q=rand(n); Q(1)=0;
Q=(1-eye(n)).*Q*0.5/mindist(Q);
r=max(min(sum(Q,2)),0.5);  
Q=diag((1:n)*2*r)-Q;
emin=min(eig(Q));
A=Q+speye(n)*(3-emin);
%%CHECK%%%
min(eig(A))
ans = 3.0000
mindist(A)
ans = 0.5000

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Answer 2

Matt J 2023-1-23

1
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1899230-matlab-problem-how-to-design-matrix-from-eigenvalues#answer_1154775

编辑：Matt J 2023-1-23

在 MATLAB Online 中打开

mindist=@(A) min(sqrt(pdist(real(A(:))).^2+pdist(imag(A(:))).^2));
n=5;
P=rand(n);
D=diag(rand(1,n));
A=P*D/P;
A=A/mindist(A)*0.5;
e=eig(A);
A=A+(3-e(1))*speye(n);
%%Check
mindist(A)
ans = 0.5000
eig(A)
ans = 5×1
    3.0000
   16.7245
   25.2124
   29.3054
   28.3224