n=3;

a=({{1, 1, 0}, {0, 1, 1}, {1, 0, 1}});a=({{1, 0, 2}, {0, 1, 1}, {2, 1, 1}})

{eval,evec}=Eigensystem[a];

evecnorm=Normalize/@evec;

m=Diagonal[Map[Reverse,Minors[a,2,Identity],{0,1}]];

evalm=Table[Eigenvalues[m[[i]]],{i,1,n}];

b=Table[Product[eval[[i]]-evalm[[j,k]],{k,1,n-1}]/Product[If[k==i,1,eval[[i]]-eval[[k]]],{k,1,n}],{i,1,n},{j,1,n}]//FullSimplify

evecnorm*Conjugate[evecnorm]//FullSimplify