Potentially nilpotent and spectrally arbitrary even cycle sign patterns Original Research Article

An n × n sign pattern image is potentially nilpotent if there is a real matrix having sign pattern image and characteristic polynomial xn. A new family of sign patterns image with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern image. These nilpotent matrices are used together with a Jacobian argument to show that image is spectrally arbitrary, i.e., there is a real matrix having sign pattern image and characteristic polynomial image for any real μi. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.