Eigentest is a package that produces real test matrices with known eigensystems. A test matrix, called an eigenmat, is generated in a factored form, in which the user can specify the eigenvalues and has some control over the condition of the eigenvalues and eigenvectors. An eigenmat A of order n requires only O(n) storage for its representation. Auxiliary programs permit the computation of (A - sI)*b, (A - sI)'*b, inv(A - sI)*b, and inv(A - sI)*b in O(n) operations. A special routine computes specified eigenvectors of an eigenmat and the condition of its eigenvalue. Thus eigenmats are suitable for testing algorithms based on Krylov sequences, as well as others based on matrix-vector products. This paper introduces the eigenmat and describes implementations in Fortran~77, Fortran~95, C, and Matlab.