In mathematics, a jacket matrix is a square symmetric matrix A= (a_{ij}) of order n if its entries are non-zero and real, complex, or from a finite field, and where In 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: 2^2 = 4, 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 N=2p If and are pxp Jacket matrix, then [A]_N 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.