如何在MATLAB中生成指定特征值的随机矩阵？是否有简便方法生成自定义特征值的随机矩阵？

Hey there! Let's break down your two MATLAB questions clearly—they're related but have key differences in how you approach them.

The go-to method here relies on similarity transformations: if two matrices are similar, they share the same eigenvalues. Here's the step-by-step approach:

1. Create a diagonal matrix D where the diagonal entries are your desired eigenvalues.
2. Generate a random invertible matrix P (orthogonal matrices work great here because their inverse is just their transpose, which is faster and numerically stable).
3. Compute your target matrix as A = P * D * P' (or P * D * inv(P) if you're not using orthogonal matrices).

Example code #

Suppose you want a 3x3 matrix with eigenvalues [2, -1, 0.5]:

% Define your desired eigenvalues
eig_vals = [2, -1, 0.5];
n = length(eig_vals);

% Generate a random orthogonal matrix (guaranteed invertible)
P = orth(randn(n));

% Diagonal matrix of eigenvalues
D = diag(eig_vals);

% Compute the random matrix with specified eigenvalues
A = P * D * P';

% Optional: Verify the result
disp("Computed eigenvalues:");
disp(sort(eig(A))); % Should match sorted eig_vals (within floating point error)

If you don't need an orthogonal matrix, you can use a random Gaussian matrix—just make sure it's invertible (check the determinant to avoid singular matrices):

P = randn(n);
% Re-generate if P is nearly singular
while abs(det(P)) < 1e-10
    P = randn(n);
end
A = P * D * inv(P);

If you want all eigenvalues to be the same (say, k), the above similarity trick with a diagonal matrix will just give you k*eye(n) (a diagonal matrix), which isn't very "random." To get a non-diagonal random matrix with identical eigenvalues, you need to use nilpotent matrices or Jordan canonical forms:

Method 1: Add a random nilpotent matrix to a diagonal matrix #

A nilpotent matrix N has the property that N^n = 0 (for an n×n matrix). Adding this to k*eye(n) gives a matrix where all eigenvalues are k (since the characteristic polynomial becomes (x - k)^n).

Example code

For a 3x3 matrix with all eigenvalues equal to 5:

k = 5;
n = 3;

% Generate a random strictly upper triangular nilpotent matrix
N = triu(randn(n), 1);

% Create the target matrix
A = k*eye(n) + N;

% Verify
disp("Eigenvalues of A:");
disp(eig(A)); % All should be ~5 (within floating point error)

Method 2: Similarity transformation on a Jordan block #

If you want a more general random matrix (not just upper triangular), you can use a Jordan block (a matrix with k on the diagonal and 1s on the superdiagonal) and apply a similarity transformation:

k = 5;
n = 3;

% Create a Jordan block with all eigenvalues k
J = k*eye(n) + diag(ones(n-1, 1), 1);

% Use a random orthogonal matrix for similarity transformation
P = orth(randn(n));
A = P * J * P';

% Verify
disp("Eigenvalues of A:");
disp(eig(A)); % All ~5

This method gives you a fully random-looking matrix where every eigenvalue is your specified value.

内容的提问来源于stack exchange，提问作者N_A