Abstract In 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 -1 AW and W -1 BW 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) T A and T B 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.
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