Given a basis $U=(u_1,\cdots,u_n)$ not necessarily orthonormal, in $\scrH$, how would you compute the biorthogonal basis $\sex{v_1,\cdots,v_n}$? Find a formula that expresses $\sef{v_j,x}$ for each $x\in\scrH$ and $j=1,\cdots,k$ in terms of Gram matrices.

Soluton. Let $V=(v_1,\cdots,v_k)$, then $$\bex V^*U=I_n\lra U^*V=I_n. \eex$$ We may just set $v_i$ to be the solution of the linear system $U^*x=e_i$, where $e_i=(\underbrace{0,\cdots,1}_{i},\cdots, 0)^T$. Suppose now $$\bex x=\sum_{j=1}^n x_jv_j\in \scrH, \eex$$ then $$\bex \sef{v_i,x}=\sum_{j=1}^n \sef{v_i,v_j}x_j,\quad i=1,\cdots,n. \eex$$ And hence $$\bex \sex{\ba{cc} \sef{v_1,x}\\ \vdots\\ \sef{v_n,x} \ea}=\sex{\sef{v_i,v_j}}\sex{\ba{cc} x_1\\\vdots\\ x_n \ea}. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]PrI.6.1