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

Hierarchy of matrix types

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 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,also it is a Diagonal block-wise inverse matrix.

Motivation

n .... -2, -1, 0 1, 2,..... logarithm
2^n .... 1, 2, 4,..... Series

As shown in Table, i.e. in series, n=2 case, Forward: , Inverse  : , then, .

Therefore, exist 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:

, and .

Therefore,

.

Also,

,.

Finally,

A·B=B·A=I

References

External links

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