Algebra multiplication

Thus, the inverse matrix plays the role of division in matrix algebra. Multiplication of equation (1.33) from the left by B and from the right by C yields... 

Another group consists of the elements 1, — 1, i and —i (where i = V i) combining operation of algebraic multiplication. 

Spectral analysis of the linearized semiflow along a periodic solution is called Floquet theory. The eigenvalues p of the linearized period map are called Floquet multipliers. A Floquet exponent is a complex / such that exp(/3r) is a Floquet multiplier of the system, where t denotes the minimal period. A periodic solution is hyperbolic if, and only if, it possesses only the trivial Floquet multiplier p = 1 on the unit circle, and this multiplier has algebraic multiplicity one. Otherwise it is called non-hyperbolic. In ODEs hyperbolic periodic solutions possess stable and unstable manifolds, similarly to the case of hyperbolic equilibria. Non-hyperbolic periodic solutions possess center manifolds. 

In matrix multiplication, a matrix times its inverse such as XX equals the identity matrix (ones on the diagonal, zero elsewhere), and in matrix algebra the identity matrix has the same role that the integer one has in algebraic multiplication which means that XX can be inserted or removed at will in a matrix equation. If we insert XX between A and B in Eq. (3.14) and premultiply and postmultiply each side of this equation hy X and X, respectively, we obtain... 

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