Higher mathematical research on the relationship between singular values and eigen values based on differential equations

In this article, the connection between eigenvalues and singular values of the linear dynamical systems is thoroughly examined in the context of differential equations. Eigenvalues have been historically applied in establishing the asymptotic stability of systems but the singular values would give an idea of the strength of the responses and temporary amplification of the system. The analysis builds a single analytical and calculational structure that combines spectral decomposition, singular value decomposition, and time-dependent analysis of a system to investigate both long-term and short-term dynamics. It is seen in the analysis that stability criteria based on eigenvalues are not enough to describe the behavior of a system fully, especially when the system is not in normal form. Numerical simulations show that even in cases where eigenvalues are in the stable region such as around −1 and −1.5, the system could still undergo considerable transient growth. Specifically, the system norm is seen to rise to an average of 2.4–6 times its original value and then decay to zero. This temporary amplification is aptly explained in singular value analysis, which is an expression of the highest energy increase of the system within limited periods of time. Additional findings indicate that the operator norm attains maximum values much larger than unity, which also establishes the short-term amplification even when the operator is stable in the long run. Also, conditioned systems are sensitive, with high condition numbers causing greater numerical instability and variability of response. The comparative study of normal and non-normal matrices underscores the fact that even though normal systems have smooth decay as the eigenvalues and their predictions, nonnormal systems exhibit significant deviations as a result of short-lived effects.