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 Λ nxn is diagonal matrix with diagonal components λ i called eigen values of matrix A U nxn is a square matrix that comprises column vectors u i called eigen vectors of matrix A An example:

The eigencomponents (every eigen value and eigen vector) are connected by linear equation (in vectors): Modeling of geochemical processes Dynamic models the example, and Determination λ i and u i It results from the former equation that ; it is

Modeling of geochemical processes Dynamic models The example: characteristic equation is rewriting gives ; solution is λ 1 = 1, λ 2 = 2 Substituting of the eigen values yields the equation set Diagonal matrix of eigen values is

Modeling of geochemical processes Dynamic models A decomposition gives - u 11 – 6 u 21 = u 11 u 11 = – 3 u 21 u 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 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 The substituting yields and It results from the second equation set The matrix of eigen vectors The equations give only direction of the eigen vectors; they give not coordinates of end point. Therefore, unitary lengths of vectors are often chosen: u u 21 2 = 1, and u u 22 2 = 1 It results from the former example u 11 2 = 9 u 21 2 and 10 u 21 2 = 1 The substituting yields and

Modeling of geochemical processes Dynamic models Verifying Inverse matrix can be find, e.g., under using of MS Excel