Matrix Science Mathematic (MSMK)

Let U={U (m,n) : m,n ∈ Z₊} _n≥m≥0 be the q-periodic discrete evolution family of square size matrices of order m having complex scalars as entries generated by L(C^m-valued, q-periodic sequence of square size matrices (A_n)_n∈Z+ where q≥2 is a natural number. Where the Poincare map U(q,0) is the generator of the discrete evolution family U. The main objective of this article to decompose C^m with the help of discrete evolution family. In fact we decompose C^m in two sub spaces X₁ and X₂ such that X₁ is due to the stability of the discrete evolution family and the vectors of X₁ will called stable vectors. While X₂ is due to the un-stability of discrete evolution family, and their vectors will be called unstable vectors. More precisely we take the dichotomy of the discrete evolution family with the help of projection P on C^m and we discuss different results of the spaces X₁ and X₂ .