Persymmetric matrix

Persymmetric matrix

In 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.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" = ("a""i"⁣"j") be an "n" × "n" matrix. The first definition of "persymmetric" requires that : a_{ij} = a_{n-j+1,n-i+1} for all "i", "j". [citation | first1=Gene H. | last1=Golub | author1-link=Gene H. Golub | first2=Charles F. | last2=Van Loan | author2-link=Charles F. Van Loan | year=1996 | title=Matrix Computations | edition=3rd | publisher=Johns Hopkins | place=Baltimore | isbn=978-0-8018-5414-9. See page 193.] For example, 5-by-5 persymmetric matrices are of the form: A = egin{bmatrix}a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \a_{21} & a_{22} & a_{23} & a_{24} & a_{14} \a_{31} & a_{32} & a_{33} & a_{23} & a_{13} \a_{41} & a_{42} & a_{32} & a_{22} & a_{12} \a_{51} & a_{41} & a_{31} & a_{21} & a_{11}end{bmatrix}.

This can be equivalently expressed as "AJ = JA"T 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.

Definition 2

The second definition is due to Thomas Muir.Muir, Thomas: "Treatise on the Theory of Determinants", page 419, Dover Press, 1960.] It says that the square matrix "A" = ("a""ij") is persymmetric if "a""ij" depends only on "i" + "j". Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form: A = egin{bmatrix}r_1 & r_2 & r_3 & cdots & r_n \r_2 & r_3 & r_4 & cdots & r_{n+1} \r_3 & r_4 & r_5 & cdots & r_{n+2} \vdots & vdots & vdots & ddots & vdots \r_n & r_{n+1} & r_{n+2} & cdots & r_{2n-1}end{bmatrix}.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.

References


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