Phased arrays are diversely applied with some specific areas including biomedical imaging and therapy, non-destructive testing, radar and sonar. In this thesis, the matrix pencil method is employed to reduce the number of elements in a linear ultrasound phased array. The noniterative, linear method begins with a specified pressure beam pattern, reduces the dimensionality of the problem, then calculates the element locations and apodization of a reduced array. Computer simulations demonstrate a close comparison between the initial array beam pattern and the reduced array beam pattern for four different linear arrays. The number of elements in a broadside-steered linear array is shown to decrease by approximately 50% with the reduced array beam pattern closely approximating the initial array beam pattern in the far-field. While the method returns a slightly tapered spacing between elements, for the arrays considered, replacing the tapered spacing with a suitably-selected uniform spacing provides very little change in the main beam and low-angle side lobes. iii ACKNOWLEDGMENT I would like to thank Dr. Robert McGough for his support and guidance during my master's degree, and for the opportunity to be part of the Biomedical Ultrasonics and Electromagnetics Lab. His experience and insight into biomedical ultrasound, phased arrays, research, and publication was an invaluable resource during my research and while writing my thesis.