On the equality of algebraic and geometric multiplicities of matrix eigenvalues
Let \(\mathbb{C}^{n\times n}\) denote the space of all complex \(n\times n\) matrices. For a matrix \(A\in\mathbb{C}^{n\times n}\), we denote the set of all eigenvalues of \(A\) by \(\sigma(A)\). Suppose \(\lambda\in\sigma(A)\). Then the algebraic multiplicity of \(\lambda\), denoted by \(\text{alg\,mult}_A(\lambda)\), is the number of times it is repeatedd as a zero of the characteristic polynomial. The geometric multiplicity of \(\lambda\), denoted by \(\text{geo\,mult}_A(\lambda)\), is the maximal number of linearly independent eigenvectors associated with \(\lambda\). In general we have \(\text{geo\,mult}_A(\lambda)\leq \text{alg\, mult}_A(\lambda)\). In this paper, the authors summarize seventeen equivalent conditions for the equality of algebraic and geometric multiplicities of an eigenvalue for a complex square matrix. As applications, new proofs of some important results related to mean ergodic and positive matrices are given.
scientific article; zbMATH DE number 5992340