Simultaneously normalizable matrices

Abstract

AbstractIn this paper two complex square matrices A and B are said to be simultaneously normalizable if there exists a nonsingular matrix W such that both W-1AW and W-1BW are normal. Beginning with some important results for one normalizable matrix, we develop a necessary and sufficient condition for two matrices A and B to be simultaneously normalizable. This condition is expressed by the properties of the modal matrices (eigenvector matrices) TA and TB of A and B. Furthermore it is shown how to detect whether the condition is fulfilled. Finally an application is demonstrated in the special case of simultaneously real symmetrizable matrices including geometrical interpretations, and a numerical example is given