[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.10

Every $k imes k$ positive matrix $A=(a_{ij})$ can be realised as a Gram matrix, i.e., vectors $x_j$, $1leq jleq k$, can be found so that $a_{ij}=sef{x_i,x_j}$ for all $i,j$.

 

Solution. By Exercise I.2.2, $A=B^*B$ for some $B$. Let $$ex B=(x_1,cdots,x_k). eex$$ Then $$ex A=sex{sef{x_i,x_j}}. eex$$