Jacket matrix

In mathematics, a jacket matrix is a square symmetric matrix formula_1 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 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: formula_5, inverse : formula_6, then, formula_7. That is, there exists an element-wise inverse.
Example 1..
or more general
Example 2..
For m x m matrices, formula_12 
formula_13
denotes an mn x mn block diagonal Jacket matrix.
Example 3..
Euler's formula:
Therefore,
Also,
Finally,
A·B = B·A = I