Persymmetric matrix
Encyclopedia
In mathematics
, persymmetric matrix may refer to:
The first definition is the most common in the recent literature. The designation "Hankel matrix" is often used for matrices satisfying the property in the second definition.
for all i, j.
For example, 5-by-5 persymmetric matrices are of the form
This can be equivalently expressed as AJ = JAT where J is the exchange matrix
.
A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes called bisymmetric matrices.
. It says that the square matrix A = (aij) is persymmetric if aij depends only on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form
A persymmetric determinant is the determinant
of a persymmetric matrix.
A matrix for which the values on each line parallel to the main diagonal are constant, is called a Toeplitz 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...
, 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.
The first definition is the most common in the recent literature. The designation "Hankel matrix" is often used for matrices satisfying the property in the second definition.
Definition 1
Let A = (aij) be an n × n matrix. The first definition of persymmetric requires that for all i, j.For example, 5-by-5 persymmetric matrices are of the form
This can be equivalently expressed as AJ = JAT where J is the 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 counterdiagonal and all other elements are zero...
.
A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes called bisymmetric matrices.
Definition 2
The second definition is due to Thomas MuirThomas Muir (mathematician)
Sir Thomas Muir FRS was a Scottish mathematician, remembered as an authority on determinants. He was born in Stonebyres in South Lanarkshire, and brought up in the small town of Biggar. At the University of Glasgow he changed his studies from classics to mathematics after advice from the future...
. It says that the square matrix A = (aij) is persymmetric if aij depends only on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form
A persymmetric determinant is the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of a persymmetric matrix.
A matrix for which the values on each line parallel to the main diagonal are constant, is called a Toeplitz matrix
Toeplitz matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...
.