New Extensions and Applications of Geršgorin Theory

In this work we discover for the first time a strong relationship between Geršgorin theory and the geometric multiplicities of eigenvalues. In fact, if λ is an eigenvalue of an n × n matrix A with geometric multiplicity k, then λ is in at least k Geršgorin discs of A. Moreover, construct the matrix C by replacing, in every row, the (k − 1) smallest off-diagonal entries in absolute value by 0, then λ is in at least k Geršgorin discs of C. We also state and prove many new applications and consequences of these results as well as we update an improve some important existing ones