  We present an eigenvalue-based approach for studying the magnetization
dynamics in magnetic nanostructures driven by spintronic excitations, such as
spin transfer torque and spin orbit torque. The approach represents the system
dynamics in terms of normal oscillation modes (eigenstates) with corresponding
complex eigenfrequencies. The dynamics is driven by a small number of active
eigenstates and often considering just a single eigenstate is sufficient. We
develop a perturbation theory that provides semi-analytical dynamic solutions
by using eigenstates for the case in the absence of damping and spintronic
excitations as a basis. The approach provides important insights into dynamics
in such systems and allows solving several difficulties in their modeling, such
as extracting the switching current in magnetic random access memories (MRAM)
and understanding switching mechanisms. We show that the presented approach
directly predicts the critical switching current, i.e., switching current for
an infinite time. The approach also provides solutions for the switching
dynamics allowing obtaining the switching current for a finite switching time,
provided that the system symmetry is broken, e.g., by tilting the polarizer, so
that switching by a finite pulse is possible.
