MATLAB Eig Function for Matrix Eigenvalue Computation

The `eig` function in MATLAB is a powerful tool for computing matrix eigenvalues and eigenvectors, widely used in scientific computing and engineering applications. Eigenvalues and eigenvectors play a crucial role in matrix analysis, revealing the intrinsic properties of matrices.

The `eig` function handles two input forms: When given a single square matrix, it returns the eigenvalues in ascending order. To obtain both eigenvalues and eigenvectors simultaneously, call `[V, D] = eig(A)` where `V` is the matrix of eigenvectors and `D` is a diagonal matrix containing eigenvalues on its main diagonal. The function automatically normalizes eigenvectors to unit length.

The function supports both real and complex matrices. Additionally, `eig` can solve generalized eigenvalue problems of the form `A*V = B*V*D` using the syntax `eig(A, B)`. For symmetric matrices, it employs specialized algorithms that ensure real-valued eigenvalues and orthogonal eigenvectors.

Underlying the `eig` function are robust numerical algorithms like the QR algorithm with implicit shifts, which ensures computational accuracy and stability through iterative refinement. For large sparse matrices, users can utilize the `eigs` function companion, which implements Arnoldi iteration methods for efficient computation of selected eigenvalues/eigenvectors without full matrix decomposition.