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Let Mn(F)Mn(F) be the space of n×nn×n matrices over a field FF. A matrix A=(A(i,j))i,j=1n∈Mn(F) is weakly symmetric if A(i,j)≠0A(i,j)≠0 iff A(j,i)≠0A(j,i)≠0 holds for all i, j . A matrix… Click to show full abstract
Let Mn(F)Mn(F) be the space of n×nn×n matrices over a field FF. A matrix A=(A(i,j))i,j=1n∈Mn(F) is weakly symmetric if A(i,j)≠0A(i,j)≠0 iff A(j,i)≠0A(j,i)≠0 holds for all i, j . A matrix is alternating if it is skew-symmetric with zero diagonal. Let Wn(F)Wn(F) and An(F)An(F) denote respectively the set of weakly symmetric matrices and the space of alternating matrices in Mn(F)Mn(F). Let [n]={1,…,n}[n]={1,…,n}. For 0≠A∈Wn(F)0≠A∈Wn(F) let q˜(A)={i,j}, where (i,j)(i,j) is the unique pair in [n]2[n]2 such that A(i,j)≠0A(i,j)≠0 and A(i′,j′)=0A(i′,j′)=0 whenever j
Journal Title: Linear Algebra and its Applications
Year Published: 2017
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