G-matrix

In linear algebra, a real invertible matrix ${\displaystyle A}$ is called a G-matrix if ${\displaystyle A^{-T}=D_{1}AD_{2}}$ (where ${\displaystyle A^{-T}}$ means ${\displaystyle (A^{-1})^{T}}$) for some real diagonal matrices ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$.

The term "G-matrix" was coined by Miroslav Fiedler and Frank J. Hall. It is sometimes called a semi-orthogonal matrix in research literature, although the latter term may also refer to a different kind of matrix, namely, a non-square matrix with orthogonal columns/rows.

All real orthogonal matrices and real invertible diagonal matrices, for instances, are G-matrices.

Properties

All matrices below are assumed to be real square matrices.

• If ${\displaystyle A}$ is a G-matrix, so are ${\displaystyle A^{T}}$ and ${\displaystyle A^{-1}}$.
• If ${\displaystyle A}$ is a G-matrix and ${\displaystyle D}$ is a nonsingular diagonal matrix, then both ${\displaystyle AD}$ and ${\displaystyle DA}$ are G-matrices.
• If ${\displaystyle A}$ is a G-matrix and ${\displaystyle P}$ is a permutation matrix, then both ${\displaystyle AP}$ and ${\displaystyle PA}$ are G-matrices.
• If ${\displaystyle A}$ is a G-matrix, then ${\displaystyle A}$ and ${\displaystyle A^{-T}}$ have the same entrywise zero pattern, i.e., ${\displaystyle A_{ij}=0}$ if and only if ${\displaystyle (A^{-T})_{ij}=0}$. Thus the entrywise zero patterns of ${\displaystyle A}$ and ${\displaystyle A^{-1}}$ are symmetric to each other.
• The direct sum of G-matrices is again a G-matrix.
• Compound matrices of a G-matrix are G-matrices.
• Kronecker products of G-matrices are G-matrices.
• Every nonsingular Cauchy matrix ${\displaystyle C}$ such that ${\displaystyle C^{-1}e}$ and ${\displaystyle C^{-T}e}$ are entrywise nonzero is a G-matrix. Here ${\displaystyle e}$ denotes the vector of ones.

References