# How to Make a Matrix Diagonal with Matlab?

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Hollis Williams on 10 Apr 2018
Edited: Walter Roberson on 1 Dec 2023
I am working on a quantum mechanics problem and would like to get a 4x4 matrix A into diagonal form such that A=UDU^{-1}. Basically I just need to know the values of D and U required in the expression to make A a diagonal matrix (where D is diagonal) as I can then use it to do an explicit calculation for a matrix exponential. As it is the matrix is not diagonal, so I cannot use the explicit expression for the matrix exponential. Is there a code with Matlab that can calculate D and U simply?
The matrix is 4 x 4 and has elements
1 0 0 1
0 -1 1 0
0 1 -1 0
1 0 0 1

Roger Stafford on 10 Apr 2018
Edited: Walter Roberson on 1 Dec 2023
or possible you need the 'eig' function at:

Cartwright on 1 Dec 2023
Edited: Walter Roberson on 1 Dec 2023
Here's an example MATLAB code: (spam link removed)
% Define your 4x4 matrix A
A = rand(4, 4); % Replace this with your actual matrix
% Compute eigenvalues and eigenvectors
[V, D] = eig(A);
% D is a diagonal matrix containing eigenvalues
% V is a matrix containing the corresponding eigenvectors
% Check if A can be diagonalized
if rank(V) == size(A, 1)
% A can be diagonalized
% Display the diagonal matrix D
disp('Diagonal matrix D:');
disp(D);
% Display the matrix of eigenvectors U
disp('Matrix of eigenvectors U:');
disp(V);
% Check the reconstruction A = U * D * inv(U)
reconstructed_A = V * D * inv(V);
% Display the reconstructed matrix A
disp('Reconstructed matrix A:');
disp(reconstructed_A);
else
disp('Matrix A cannot be diagonalized');
end
Replace the rand(4, 4) with your actual 4x4 matrix A. The code uses eig to compute the eigenvalues (D) and eigenvectors (V). It then checks if A can be diagonalized by verifying that the rank of the matrix of eigenvectors is equal to the size of the matrix. If it can be diagonalized, it displays the diagonal matrix D and the matrix of eigenvectors U.

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