All physical transformations: reflection, translation, rotation, dilation, and contraction, etc. can be accomplished simply by the use of special matrices. Multiplying them result the desired transformations. Although transformations can be achieved by scalar multiplication, for the sake of clarity, they are best done by matrix multiplication. On the other hand, if bases are defined then linear transformations of vector spaces with respect to these bases can also be accomplished. Clearly, there exist two distinct types of transformation: matrix multiplication and matrix addition. Only the latter requires the definition of spanning bases for specifying a given coordinate system.
Nonetheless, the special forms of singular Hadamard matrices can be used to achieve both physical mass development by matrix multiplication and fractional electric charge development by matrix addition. Each double spin angular transformation can be used to define spacetime charges of H-plus and H-minus. These unique transformations completely spanned the totality of three dimensional spatial components of the spacetime continuum. Consequently, the time component becomes the continuous infinitesimal local motion of these angular transformations. Hence, one Hadamard matrix (either H-plus or H-minus) is more than sufficient to represent an angular matrix for the justification of a principle of double least action at the subquantum domain of physical reality.


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