Jacket matrix

In mathematics, a jacket matrix is a square symmetric matrix of order n if its entries are non-zero and real, complex, or from a finite field, and
where I_n is the identity matrix, and
where T denotes the transpose of the matrix.
In other words, the inverse of a jacket matrix is determined by its element-wise or block-wise inverse. The definition above may also be expressed as:
The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.

Motivation

As shown in the table, i.e. in the series, for example with n=2, forward:, inverse :, then,. That is, there exists an element-wise inverse.

Example 1.

or more general

Example 2.

For m x m matrices,
denotes an mn x mn block diagonal Jacket matrix.

Example 3.

Euler's formula:
Therefore,
Also,
Finally,
A·'B = B'·A = '''I'''

Example 4.

Consider be 2x2 block matrices of order
If and are pxp Jacket matrix, then is a block circulant matrix if and only if, where rt denotes the reciprocal transpose.

Example 5.

Let and, then the matrix is given by
where U, C, A, G denotes the amount of the DNA nucleobases and the matrix is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.