Exchange matrix

Exchange matrix

In mathematics, especially linear algebra, the exchange matrix is a special case of a permutation matrix, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix.

:J_{2}=egin{pmatrix}0 & 1 \1 & 0end{pmatrix};quad J_{3}=egin{pmatrix}0 & 0 & 1 \0 & 1 & 0 \1 & 0 & 0end{pmatrix};quad J_{n}=egin{pmatrix} 0 & 0 & cdots & 0 & 0 & 1 \ 0 & 0 & cdots & 0 & 1 & 0 \ 0 & 0 & cdots & 1 & 0 & 0 \ vdots & vdots & & vdots & vdots & vdots \ 0 & 1 & cdots & 0 & 0 & 0 \ 1 & 0 & cdots & 0 & 0 & 0 end{pmatrix}.

Definition

If "J" is an "n×n" exchange matrix, then the elements of "J" are defined such that::J_{i,j} = egin{cases} 1, & j = n - i \0, & j e n - i \end{cases}

Properties

* "J"T = "J".
* "Jn" = "I" for even "n"; "Jn" = "J" for odd "n", where "n" is any integer. Thus "J" is an involutary matrix; that is, "J"−1 = "J".
* The trace of "J" is "1" if "n" is odd, and "0" if "n" is even.

Relationships

* Any matrix "A" satisfying the condition "AJ = JA" is said to be centrosymmetric.
* Any matrix "A" satisfying the condition "AJ = JA"T is said to be persymmetric.


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