Matrices and Determinants

In terms of the Hamilton formalism a hypercomplex number of unit norm can now be defined in the form 

These numbers do not obey all of the laws of the algebra of complex numbers. They add like complex numbers, but their multiplication is not commutative. The general rules of multiplication of n-dimensional hypercomplex numbers were investigated by Grassmann who found a number of laws of multiplication, including Hamilton s rule. These methods still await full implimentation in physical theory. 

The first successful formulation of quantum mechanics by Heisenberg was achieved in terms of matrices, and has not diminished in importance since 

To reproduce the system of equations it is noted that each row of the matrix is multiplied into the column vector, adding elementary products together to yield the elements of the product vector, b. This is a special case of matrix multiplication which can be represented symbolically by 

The zth row of the pre-factor is multiplied term-by-term with the jth column of the post-factor and added to yield the ijth element of the product. 

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Aitken, A. C, Determinants and Matrices, John Wiley Sons, New York (1956). 

Determinants and Matrices STATl ST 1CA L M A T11E MATICS Waves Electricity Projective Geometry Integration... 

A.C. Aitken, Determinants and Matrices, (McGraw-Hill Book Company, New York, 1968). 

M— DETERMINANTS AND MATRICES. 1. The Formation and Expansion of Determinants, pages 377-380. 2. Minor Determinants, pages 380-383. 3. Differentiation of a Determinant, pages 383-384. 4. General... 

H. T. Brown, Introduction to the Theory of Determinants and Matrices, University of North Carolina Press, Chapel Hill, N.C., 1958... 

Because this process requires a httle knowledge of linear algebra (determinants and matrices), a limited review is given in Appendix A (matrix algebra). Here, we introduce the concept through a simple example. 

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