G-matrix

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

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 $A$ is a G-matrix, so are $A^{T}$ and $A^{-1}$.
• If $A$ is a G-matrix and $D$ is a nonsingular diagonal matrix, then both $AD$ and $DA$ are G-matrices.
• If $A$ is a G-matrix and $P$ is a permutation matrix, then both $AP$ and $PA$ are G-matrices.
• If $A$ is a G-matrix, then $A$ and $A^{-T}$ have the same entrywise zero pattern, i.e., $A_{ij}=0$ if and only if $(A^{-T})_{ij}=0$. Thus the entrywise zero patterns of $A$ and $A^{-1}$ 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 $C$ such that $C^{-1}e$ and $C^{-T}e$ are entrywise nonzero is a G-matrix. Here $e$ denotes the vector of ones.