Exchange matrix
Encyclopedia
In mathematics
, especially linear algebra
, the exchange matrix is a special case of a permutation matrix
, where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the exchange matrix is a special case of a permutation matrix
Permutation matrix
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
, where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
.
Definition
If J is an n×n exchange matrix, then the elements of J are defined such that:Properties
- JT = 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 traceTrace (linear algebra)In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of J is 1 if n is oddEven and odd numbersIn mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
, and 0 if n is evenEven and odd numbersIn mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
.
Relationships
- Any matrix A satisfying the condition AJ = JA is said to be centrosymmetricCentrosymmetric matrixIn mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n × n matrix A = [ Ai,j ] is centrosymmetric when its entries satisfy...
. - Any matrix A satisfying the condition AJ = JAT is said to be persymmetricPersymmetric matrixIn mathematics, persymmetric matrix may refer to:# a square matrix which is symmetric in the northeast-to-southwest diagonal; or# a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line....
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