Bisymmetric matrix

Symmetry pattern of a bisymmetric (5 × 5) matrix

In mathematics, a bisymmetrical matrix or doubly symmetrical matrix is a square matrix that is symmetrical both with regard to its main diagonal and with regard to its opposite diagonal .

definition

A square matrix over a body is called bisymmetric if for its entries ${\ displaystyle A \ in K ^ {n \ times n}}$ ${\ displaystyle K}$

{\ displaystyle a_ {i, j} = a_ {j, i}}

and

{\ displaystyle a_ {i, j} = a_ {n-j + 1, n-i + 1}}

for applies. The entries of a bisymmetric matrix do not change if they are mirrored on the main diagonal or on the opposite diagonal. ${\ displaystyle i, j = 1, \ ldots, n}$

Examples

Bisymmetric matrices of size have the general shape ${\ displaystyle 3 \ times 3}$

{\ displaystyle A = {\ begin {pmatrix} a & b & c \\ b & d & b \\ c & b & a \ end {pmatrix}}}

and those of size the shape ${\ displaystyle 4 \ times 4}$

{\ displaystyle A = {\ begin {pmatrix} a & b & c & d \\ b & e & f & c \\ c & f & e & b \\ d & c & b & a \ end {pmatrix}}}

with . ${\ displaystyle a, b, c, d, e, f \ in K}$

properties

Symmetries

A bisymmetrical matrix is both symmetrical and persymmetrical and thus also centrally symmetrical . Conversely, a centrally symmetric matrix, which is also symmetric or persymmetric, is bisymmetric. With the permutation matrix defined by ${\ displaystyle J \ in K ^ {n \ times n}}$

{\ displaystyle J = (\ delta _ {i, n-j + 1}) _ {ij} = {\ begin {pmatrix} 0 && 1 \\ & \ cdot ^ {\, \ cdot ^ {\, \ cdot}} & \\ 1 && 0 \ end {pmatrix}}}

bisymmetric matrices can also be made compact by the two conditions

{\ displaystyle A = A ^ {T}}

and

{\ displaystyle JA = AJ}

characterize. A real symmetric matrix is bisymmetric if and only if its eigenvalues differ from left or right after multiplication with the matrix at most with respect to the sign . ${\ displaystyle J}$

Sum and product

The sum of two bisymmetric matrices and again results in a bisymmetric matrix, also scalar multiples are with . Since the zero matrix is trivially bisymmetric, the bisymmetric matrices form a sub-vector space in the matrix space . ${\ displaystyle A + B}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle cA}$ ${\ displaystyle c \ in K}$ ${\ displaystyle K ^ {n \ times n}}$

The product of two bisymmetric matrices results in a bisymmetric matrix again if the two matrices and commute . ${\ displaystyle A \ cdot B}$ ${\ displaystyle A}$ ${\ displaystyle B}$

Inverse

For the inverse of a bisymmetric matrix, if it exists ${\ displaystyle A ^ {- 1}}$

{\ displaystyle A ^ {- 1} = (A ^ {T}) ^ {- 1} = (A ^ {- 1}) ^ {T}}

and .

{\ displaystyle YES ^ {- 1} = (AJ) ^ {- 1} = (YES) ^ {- 1} = A ^ {- 1} J}

The inverse of a regular bisymmetric matrix is therefore again bisymmetric.

Individual evidence

^ Thomas Muir: A Treatise on the Theory of Determinants . Dover, New York 1960, pp. 19 .
↑ David Tao, Mark Yasuda: A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices . In: SIAM J. Matrix Anal. Appl . tape 23 , no. 3 , 2002, p. 885-895 .
^ Gene Golub, Charles van Loan: Matrix Computations . JHU Press, 2013, p. 208 .

Web links

Eric W. Weisstein : Bisymmetric Matrix . In: MathWorld (English).

[1] Thomas Muir: A Treatise on the Theory of Determinants . Dover, New York 1960, pp. 19 .

[2] David Tao, Mark Yasuda: A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices . In: SIAM J. Matrix Anal. Appl . tape 23 , no. 3 , 2002, p. 885-895 .

[3] Gene Golub, Charles van Loan: Matrix Computations . JHU Press, 2013, p. 208 .