It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical applications not only in linear algebra itself but also in more technical disciplines such as, for example, the study of symmetry groups in crystallography. This article may complement the usual material found in textbooks and whet the students' appetite for advanced topics in linear algebra. In this note, the author starts with a basis which is not necessarily orthonormal and derives a formula for the matrix which possesses members of this basis as eigenvectors. He discusses some problems such as the following: (1) Is it possible for distinct matrices to have the same set of eigenvalues and eigenvectors?; and (2) It is well-known that a symmetric matrix of order "n" possesses a set of "n" orthonormal eigenvectors. Does this mean that if a matrix is "not" symmetric then this property no longer holds? The author also gives a simple proof of the spectral decomposition theorem.