Euler Hadamard/DCT polynomial matrix

Abstract

The Euler Hadamard/DCT polynomial is defined in this paper. This polynomial is similar to the Euler theorem in that it calculates the unit operation. However, the Euler Hadamard/DCT polynomial is computed by using matrix operations and angle information. The computations of Euler Hadamard/DCT polynomial can be used to construct higher order of Hadamard/DCT matrices and other real orthogonal matrices. Specially, the inverses of these Euler Hadamard polynomials are simply from the element inverse and the basic idea is corresponding to the polynomial function X N ·( X N ) T = NI N with Hadamard computations [ H] N ·([ H] N ) T = N[ I] N , which is the unit operation of orthogonal matrix. Otherwise, from the geometric view, we give a briefly description to the Euler Hadamard/DCT polynomial. The geometric structure shows that there possibly exist some other orthogonal or element-wise inverse matrices (or polynomials) by using the generalized Euler Hadamard/DCT polynomial.