Let A = I-T, where T is any stochastic matrix and let E be a perturbation matrix such that T-E is a stochastic matrix.

Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … The group inverse of A is denoted by A^♯. In addition, the technique may be used to convert any non-singular matrix into a singular matrix by replacing any one or several entries in the original matrix.

Formulation.

Note that the roots of characteristic polynomials are eigenvalues. ... We prove that a perturbation of a singular matrix is nonsingular.

For a given n × n non-singular matrix A, its inverse matrix A − 1 is first evaluated. Does someone know theorems about approximating the inverse of a matrix through perturbation theory?I would be very grateful, if you could recommend me some literature on that. A typical example is provided to show the merit of the approach presented. In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. 2. We prove that a perturbation of a singular matrix is nonsingular. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues are to changes in the system.