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. Note that the roots of characteristic polynomials are eigenvalues. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues are to changes in the system. Formulation. We prove that a perturbation of a singular matrix is nonsingular. The group inverse of A is denoted by A^♯. For a given n × n non-singular matrix A, its inverse matrix A − 1 is first evaluated. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 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. 2. A typical example is provided to show the merit of the approach presented.

... We prove that a perturbation of a singular matrix is nonsingular. 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.

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.