Jacobi Method for Computing Matrix Eigenvalues and Eigenvectors

The Jacobi method is a classical numerical algorithm designed to compute all eigenvalues and corresponding eigenvectors of real symmetric matrices. Its core principle involves diagonalizing the matrix through a series of orthogonal similarity transformations (Jacobi rotations), ultimately revealing the eigenvalues and eigenvectors.

Methodology Overview The Jacobi method operates iteratively by selecting the off-diagonal element with the largest absolute value in each step. A rotation matrix is constructed to annihilate this element, progressively driving the matrix toward diagonal form. Through repeated iterations, off-diagonal elements approach zero, at which point the diagonal elements become the eigenvalues, and the columns of the accumulated product of rotation matrices form the eigenvectors.

MATLAB Implementation Key implementation steps in MATLAB include: Initialization: Verify the input matrix is real symmetric (using issymmetric() function), as the method fails otherwise. Iteration process: For each iteration, locate the maximum off-diagonal element (using max() and abs() functions), compute the rotation angle θ = 0.5*atan(2*a_ij/(a_ii-a_jj)), construct the Givens rotation matrix, and update the current matrix via A = Q'*A*Q. Convergence check: Terminate when all off-diagonal elements fall below a predefined tolerance (e.g., 1e-10) using norm(A,'fro')-norm(diag(A)) calculations.

Debugging and Optimization Critical considerations during implementation: Numerical stability: Address floating-point precision issues by implementing threshold-adaptive convergence conditions and avoiding catastrophic cancellation in angle calculations. Efficiency optimization: Reduce O(n³) complexity for large matrices by implementing threshold-based element selection and exploiting matrix sparsity where applicable.

Application Scenarios The Jacobi method suits small-to-medium real symmetric eigenvalue problems, including vibration analysis in physics, principal component analysis (PCA) in statistics, and structural mechanics computations.

Successful MATLAB implementation validates algorithmic correctness through comparison with eig() function results, providing a foundation for advanced matrix computations like singular value decomposition.