Pearce element ratios: A linear algebra perspective.!! Input to mass-transfer problems

Transcription

1 Pearce element ratios: A linear algebra perspective.!! Terry Gordon University of Calgary University of British Columbia Input to mass-transfer problems Geochemical matrices (GCM) of chemical analyses Mass transfer matrices (MTM) of hypothesized processes - compositions added and/or subtracted to protolith fo fa dihd ab an 3 Ti Al Mn Ca Na K P

2 Conventions m rows (elements) m- dimensional space Ti Al n columns (analyses) # # # 3 # 4 m x n matrix of measurements Theoretical compositions as molecular formulae! Why examine algebraic properties of mass-transfer matrix? see strength of the hypothesis - will it be consistent with many compositions (weak) or only a few compositions (strong). provide a basis for selecting element ratios applicable to the hypothesis.

3 Example MTM and vectors fa fo Matrix rank fa fo # # x 4 matrix, but any column vector can be expressed as a combination of any other two. # =.6 fa +.4 fo fa = -3/7 fo + /7 # The matrix rank is. 3

4 Rank tells us the number of independent processes modeled by the MTM In this example there are two independent directions, so possible processes lie on a -d plane. In a real problem ( + elements) rank is not so obvious. Rank also tells us about the number of invariant chemical vectors modeled by the MTM Invariant means places the process can t go. e.g. a conserved or immobile element or a ratio of elements. The number of independent invariants is (m - rank) 4

5 Orthogonality ANY two of the column vectors in the example MTM define all possible masstransfer vectors. The remaining space, invariant under mass-transfer, fills up 3d space, hence is onedimensional - a vector. It is orthogonal to every vector defined by the MTM. In spaces of greater dimension there may be more than one. Orthogonality Vectors are orthogonal if their inner product is zero. Inner product is computed by multiplying corresponding terms and summing the results. In the example, the inner product of the invariant vector [ - -] with [ ] and [ ] and all their combinations is zero. This generalizes to = 5

6 Orthogonality 3 The relationship - - = can be expressed in several different ways. Rearranging gives ( + ) =. This shows that a plot of ( + ) versus will be a straight line with slope. Alternatively ( + )/ = is an invariant ratio. Computing the rank and a set of independent invariant vectors If A is a mass-transfer matrix, Matlab provides a simple method of determining these values. >> RankA = rank(a)! >> Invars = null(a, r )!! Note that invariant vectors include not only those returned by the program, but any linear combination of them. 6

7 Na K Some examples olivine sorting MTM! MTI! olivine sorting, Na mobile Na K olivine sorting, Na-K exchange Na K More examples MTM! MTI! Assimilation single composition Na K

8 Summary Models of mass-transfer process can be expressed as m n matrices. Algebraic analysis of these matrices will reveal the rank - number of independent directions mass transfer can take place. The analysis can also reveal a set of invariant vectors. These vectors and/or their linear combinations can be used to determine invariant ratios and axes on PER diagrams. 8

9 Vector and matrix multiplication Multiplying a vector or matrix by a constant is straightforward. Multiply each term by the constant.! 5 = 5!!!! = The dot or inner product of two vectors is computed by multiplying corresponding terms and summing the result. This produces a scalar - a single number. Obviously the vectors must have the same number of terms. 4 times = 3 = 5 This is usually written as the transpose of the first vector times the second. This is a row times a column or across times down - an easy way to remember. 4 3 = (4 ) + ( 3) + ( 4) = 8 3 = 5 4 A very useful fact, that we wonʼt prove, is that if two vectors are orthogonal - at right angles - their dot product is zero. Multiplication of two matrices (A*B) is performed by computing the dot product of each row in matrix A with each column in matrix B. The result goes into the corresponding row and column of the product matrix C. So - across times down and save in same row and column = Second row in A times second column in B equals (,) entry in C. 3 = ( ) + (3 3) = 9 3 Obviously the number of columns in the first matrix must equal the number of rows in the second matrix. The product has the same number of rows as the first matrix and the same number of columns as the second matrix. m n n p = m p

10 Vector and matrix multiplication These definitions means that there is another vector product - the outer product. This is a column vector times a row vector. A*B = C becomes 3 = which certainly satisfies the columns in first = rows in second rule. The result is a one-dimensional matrix. Each column in the m x n product C is a multiple of the m x column matrix A. The multipliers are the entries in the x n matrix B.

11 Resources RREF online SVD online RREF for Excel download "Excel Pivot and Gauss-Jordan Tool" SVD for Excel unknown Free Matlab-like software Octave Scilab Commercial software Matlab References Strang, G. Introduction to Linear Algebra. Any edition. You can get the latest directly from: The prices of used copies of Professor Strang's books reflect the fact that few owners wish to sell. itunes U. Professor Strang's lectures. Teaching codes to accompany Professor Strang's text.