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Unbalanced phase voltages in electricity networks prevent efficient and stable delivery of power and energy services from utility companies to end-users, and vice versa. While variations in voltage magnitude and power loss have been studied extensively, little attention has been paid to the development of linearized expressions to account for the variation in voltage unbalance. Using a standard definition of the voltage unbalance factor (VUF), this study is the first, to the best knowledge of the authors, in which a VUF sensitivity matrix is mathematically developed and analyzed with respect to variations in 3-phase (3-ph) voltages and, consequently, 3-ph unbalanced load demand. To demonstrate its utility, we performed simulation case studies and compared the estimates made using the matrix to those from a power flow calculation algorithm. The VUF sensitivity matrix is expected to have a variety of practical applications for power systems, such as optimal scheduling and real-time control, which enables improvement in network power quality.