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Jacket matrix

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Jacket matrix

In mathematics, a jacket matrix is a square symmetric matrix A = ( a i j ) of order n if its entries are non-zero and real, complex, or from a finite field, and

Contents

  A B = B A = I n

where In is the identity matrix, and

  B = 1 n ( a i j − 1 ) T .

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:

∀ u , v ∈ { 1 , 2 , … , n } :   a i u , a i v ≠ 0 ,         ∑ i = 1 n a i u − 1 a i v = { n , u = v 0 , u ≠ v

The jacket matrix is a generalization of the Hadamard matrix,also it is a Diagonal block-wise inverse matrix.

Motivation

As shown in Table, i.e. in series, n=2 case, Forward: 2 2 = 4 , Inverse  : ( 2 2 ) − 1 = 1 4 , then, 4 ∗ 1 4 = 1 .

Therefore, exist an element-wise inverse.

Example 1.

A = [ 1 1 1 1 1 − 2 2 − 1 1 2 − 2 − 1 1 − 1 − 1 1 ] , : B = 1 4 [ 1 1 1 1 1 − 1 2 1 2 − 1 1 1 2 − 1 2 − 1 1 − 1 − 1 1 ] .

or more general

A = [ a b b a b − c c − b b c − c − b a − b − b a ] , : B = 1 4 [ 1 a 1 b 1 b 1 a 1 b − 1 c 1 c − 1 b 1 b 1 c − 1 c − 1 b 1 a − 1 b − 1 b 1 a ] ,

Example 2.

For m x m matrices, A j ,

A j = d i a g ( A 1 , A 2 , . . A n ) denotes an mn x mn block diagonal Jacket matrix.

J 4 = [ I 2 0 0 0 0 c o s θ − s i n θ 0 0 s i n θ c o s θ 0 0 0 0 I 2 ] ,   J 4 T J 4 = J 4 J 4 T = I 4 .

Example 3.

Euler's Formula:

e i π + 1 = 0 , e i π = cos ⁡ π + i sin ⁡ π = − 1 and e − i π = cos ⁡ π − i sin ⁡ π = − 1 .

Therefore,

e i π e − i π = ( − 1 ) ( 1 − 1 ) = 1 .

Also,

y = e x

d y d x = e x , d y d x d x d y = e x 1 e x = 1 .

Finally,

A·B=B·A=I

References

Jacket matrix Wikipedia
(Text) CC BY-SA


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