Sincerely,

Clint Holt

12-12-09

 

Several years ago, quite by accident, I discovered an interesting property of matrices that has been overlooked in the mainstream of Mathematics. If we have two conformable matrices, transpose the pre-multiplier, take the columns of the transposed pre-multiplier, in order, and multiply them straight across the post-multiplier matrix (as we do Gausian Reduction), we obtain a set of half-multiplied matrices, or nested arrays. If the pre-multiplier was a row matrix, this half-multiplication gives us a new matrix, if the pre-multiplier was a matrix, we obtain a nested array. I call this the Half-Multiplier because, unlike in regular matrix multiplication where we multiply and sum, with this operator we do the multiplication but do not necessarily sum the products. If we transpose the resulting nested array and sum the columns, we acheive the same results as in regular matrix multiplication. If we sum the rows instead of the columns, we acheive the Cross Product of a matrix (as of yet undefined in modern mathematics). If we add the individual matrices (sub-matrices) of the nested array together, we achieve the Matric Product of a matrix. Also, under Half-Multiplication, matrices become transpose cummutative. This is important because what was believed to be a non-commutative operator is now commutative. This operator seems to generate and govern nested arrays.
With this operator, just by adding a constant term, there seems to be generated a generalized field equation (the Mother Equation) from which local field equations may be formulated. Ie. , we can derive the field equations for Classical Physics, Quantum Mechanics, Astro-physics, Accounting/Inventory systems and with a little "twist" can derive a field equation for statistics which can be derived for all the above field equations. These equations all have an added term not found in the known field equations which, if the added term is the identity matrix, the equations all reduce to the field equations in their familiar forms with their connection to statistics.
Because this operator is new, no known math packages can handle it. But fortunately it has equivalent operations in regular matrix math which can be substituted. I.e. Multiplying by a diagonal matrix gives the same solution as half-multiplying. Also, since no programs can handle nested arrays, we overcome this difficulty by writing the nested array as a single array and multiply as needed to achieve the same results as multiplying by a nested array.
One final note, a number can be considered to be a tensor of rank zero. Reversing this, if a tensor of any rank can be considered to be a single number, then the math in this book reduces to simple arithmetic. The new "number" has all the properties of single numbers except we do all operations all at the same time. This goes contrary to the modern assumption that there is only one way to multiply a matrix and that there is no element by element multiplication of matrices. I have offered a proof for this as well as offer a proof for Gaussian Reduction in this book.

 

The downloads on the right are in order, pg. 1-108, pg. 109-218, pg. 219-330,  pg. 331-440,  pg. 441-550,  pg. 551-660,  pg. 661-727.  The Table of Contents are not close and may not lead you to the exact page, but they will get you close, at least within 40 pages.

 

The last link (or link # 8) contains just the proofs, examples of how to multiply nested arrays using regular matrix methods (until computers are programmed to compute nested arrays anyway), a  3-variable Analysis of Variance statistical example and a Chemical Usage Inventory.  Environmental Managers are welcome to use this for their TRI reports, however, consultants who need the statistical analysis of the chemical data and trend analysis, please e-mail me for more info.

 

I have finally gotten back online after 5 years (5-14-07).  I am about to publish this book as a Publish on Demand copy.  It will still be several month before it is ready, and it will probably be offered on Amazon.com.  I have found a 7 step method of writing analysis of Variance into a single equation that works for all AOV if the mathematical processes are the same, two Mathematical sentences if we have both Between and Within Subjects computations, but the two can still be summed into a single sencence (expression).  The publisher demanded the book be shortened, so I have cut down on the easy Statistics and concentrated on the harder, more advanced problems.  Please let Amazon know if you are interested.