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Modeling of geochemical processes Linear A lgebra Refreshing

Modeling of geochemical processes Linear A lgebra 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.

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Modeling of geochemical processes Linear A lgebra Refreshing

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  1. Modeling of geochemical processesLinear Algebra Refreshing J. Faimon

  2. Modeling of geochemical processesDynamic models Eigen Values and Eigen Vectors of Matrix The square matrix Anxn can be decomposed (factorized) into a product of three matrix Λnxn is diagonal matrix with diagonal components λi called eigen values of matrix A Unxn is a square matrix that comprises column vectors ui called eigen vectors of matrix A An example:

  3. Modeling of geochemical processesDynamic models The eigencomponents (every eigen value and eigen vector) are connected by linear equation (in vectors): the example , and Determination λi andui It results from the former equation that ; it is

  4. Modeling of geochemical processesDynamic models The example: characteristic equation is rewriting gives ; solution is λ1 = 1, λ2 = 2 Diagonal matrix of eigen values is Substituting of the eigen values yields the equation set

  5. Modeling of geochemical processesDynamic models A decomposition gives - u11 – 6 u21 = u11 u11 = – 3 u21 u11 + 4 u21 = u21 u11 = – 3 u21both equations are depended and - u12 – 6 u22 = 2 u12 u12 = – 2 u22 u12 + 4 u22 = 2 u22 u12 = – 2 u22 both equations are depended If it is chosen u21 = -1, then u11 = 3. If u22 = -1, then u12 = 2!

  6. Modeling of geochemical processesDynamic models The equations give only direction of the eigen vectors; they give not coordinates of end point. Therefore, unitary lengths of vectors are often chosen: u112 + u212 = 1, and u122 + u222 = 1 It results from the former example u112 = 9 u212 and 10 u212 = 1 The substituting yields and It results from the second equation set The substituting yields and The matrix of eigen vectors

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

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