| singular nature
Square symmetric Hadamard matrices are known to be singular that is their determinants are all zeros. However, it can be shown that the sum of 2 singular matrices may be nonsingular and vice versa. Moreover, it is known in linear algebra that the product of 2 square matrices is singular if and only if at least one of the matrices is singular.
If a, b, and c are Pauli spin anticommutative nonsingular 2 by 2 spin matrices then it can be shown that the 2 by 2 Hadamard matrix H(2)+ = a-a and H(2)- = a-a. If a, b, g, and d are the 4 by 4 Dirac anticommutative matrices then the 4 by 4 Hadamard matrix H(4)+ = a-a+m+n and H(4)- = a-a+m+n where m and n are both trace zero commutative symmetric self-inverse nonsingular square matrices of unit determinant. The only exceptions are the zero matrices. The advantages of Hadamard matrices are real elements and commutative while Pauli and Dirac matrices contain imaginary numbers and anticommutative.
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Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
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Linear Mode
