zero power matrix

[AntonioLao]

Certain definitions in mathematics give peculiarities that defy complete understanding before one attempts of becoming a PhD of mathematics. One example is the zero power of a matrix. Given a matrix A and its inverse A¯¹, their product is the identity matrix: A A¯¹ =A¯¹A = I. In the real number system, this is the same as dividing a number by itself to give unity, for example, 2 ÷ 2 = 1 or �� ÷ �� = 1. In these cases, the meaning of the identity matrix is equivalent to the unity of real numbers wherein it can be understood that the inverse of a matrix is the same as the reciprocal of the matrix: A¯¹ = 1/A or A ÷ A = I. In the imaginary complex domain, the same definitions can apply. However, the inverse of imaginary unity is the negative of imaginary unity: (��)(-��) = ��÷�� = 1 while the inverse of an arbitrary complex number (��+����) is (��-����)/(��²+��²) such that their product is unity. Although all real and complex imaginary numbers have inverses, not all matrices have inverses. In particular, the square symmetric Hadamard matrices have no inverses. Nevertheless, the zero power of any square symmetric Hadamard matrix is equal to the identity matrix of the same order: A⁰ = I, where A has no inverse.