This paper investigates the focusing properties of linear magnetic undulators, i.e., devices characterized by weak defocusing properties in the horizontal (wiggling) plane and strongly focusing in the vertical plane. The problem of identifying the conditions that ensure the existence of the electron beam eigenstates in the undulator lattice for a given working point of electron beam energy Eb and resonant wavelengthr is studied. For any given undulator lattice, a bandlike structure is identified defining regions in the ðEb ;� rÞ plane where no periodic matching condition can be found, i.e., it is not possible to transport the electron beam so that optical functions are periodic at lattice boundaries. Some specific cases are discussed for the SPARC FEL undulator. I. INTRODUCTION Pure permanent magnet (PPM) undulators are imple- mented in single pass free-electron lasers (FEL) as inser- tion devices featuring a magnetostatic periodic (sinusoidal) field in the longitudinal (z) direction. The transverse magnetic field ensures coupling between the transverse wiggling motion of electrons and the transverse field of the amplified optical wave. The FEL power gain length, i.e., the exponential growth folding length, is Lg ¼ � u=ð4�� ffiffiffi p Þ, where � u is the undulator period andis the Pierce parameter (1-3), proportional to the cubic root of the current density. An FEL amplifier operating in self- amplified spontaneous emission has a saturation length Ls � 18-20Lg. We typically have Ls ranging from few meters in long wavelength (visible to near UV) devices, up to several tens of meters, in hard x-ray FEL amplifiers. The optimization process of the FEL parameters has to take into consideration the focusing properties of the undulator and the design of the electron transport lattice over the undulator length, in order to optimize the transverse beam size and maximize the current density, reducing therefore the gain length. The transport lattice of FEL amplifiers operating with high energy beams, as typical soft to hard x-ray sources (4-7), is based on a focusing-drift- defocusing-drift (FODO) scheme, alternated to a seg- mented undulator (8). The effect of a (linear) undulator is that of introducing a small perturbation to the FODO: a weak focusing in the vertical plane and a negligible effect in the horizontal plane. At lower beam energies, i.e., for FEL amplifiers operating at longer wavelengths from vis- ible to UV, the role of the undulator focusing becomes