Modeling of geochemical processes
                                      Linear Algebra Refreshing

                                             J. Faimon

                                 Modeling of geochemical processes
                                           Dynamic models

 Eigen Values and Eigen Vectors of Matrix 

The square matrix A[nxn] can be decomposed (factorized) into a product of three matrix

                                 Modeling of geochemical processes
                                           Dynamic models

The eigencomponents (every eigen value and eigen vector) are connected by linear equation (in
vectors):

                                 Modeling of geochemical processes
                                           Dynamic models

The example:

                                 Modeling of geochemical processes
                                           Dynamic models

A decomposition gives      

  - u[11] -- 6 u[21]  =  u[11]                 u[11] =  -- 3 u[21]

    u[11] + 4 u[21]  =  u[21]                 u[11] =  -- 3 u[21]        both equations are depended

and              

  - u[12] -- 6 u[22]  =  2 u[12]              u[12] =  -- 2 u[22]

    u[12] + 4 u[22]  =  2 u[22]              u[12] =  -- 2 u[22]        both equations are depended



If it is chosen   u[21] = -1, then u[11] = 3.     If  u[22] = -1, then u[12] = 2!



                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

Verifying