1 problem found
Let \(a,b,c,d,p,q,r\) and \(s\) be real numbers. By considering the determinant of the matrix product \[ \begin{pmatrix}z_{1} & z_{2}\\ -z_{2}^{*} & z_{1}^{*} \end{pmatrix}\begin{pmatrix}z_{3} & z_{4}\\ -z_{4}^{*} & z_{3}^{*} \end{pmatrix}, \] where \(z_{1},z_{2},z_{3}\) and \(z_{4}\) are suitably chosen complex numbers, find expressions \(L_{1},L_{2},L_{3}\) and \(L_{4},\) each of which is linear in \(a,b,c\) and \(d\) and also linear in \(p,q,r\) and \(s,\) such that \[ (a^{2}+b^{2}+c^{2}+d^{2})(p^{2}+q^{2}+r^{2}+s^{2})=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+L_{4}^{2}. \]
Solution: Supppose \(z_1 = a+ib, z_2 = c+id, z_3 = p+iq, z_4 = r+is\) then: \begin{align*} && \det \left (\begin{pmatrix}z_{1} & z_{2}\\ -z_{2}^{*} & z_{1}^{*} \end{pmatrix}\begin{pmatrix}z_{3} & z_{4}\\ -z_{4}^{*} & z_{3}^{*} \end{pmatrix} \right) &= \det \begin{pmatrix}z_{1} & z_{2}\\ -z_{2}^{*} & z_{1}^{*} \end{pmatrix}\det\begin{pmatrix}z_{3} & z_{4}\\ -z_{4}^{*} & z_{3}^{*} \end{pmatrix} \\ && \det \begin{pmatrix}z_{1}z_3-z_2z_4^* & z_1z_4+z_2z_3^*\\ -z_2^*z_3-z_1^*z_4*& -z_2^*z_4+z_{1}^*z_3^* \end{pmatrix}&= (z_1z_1^*+z_2z_2^*)(z_3z_3^*+z_4z_4^*) \\ && |z_{1}z_3-z_2z_4^*|^2+|z_1z_4+z_2z_3^*|^2&= (a^2+b^2+c^2+d^2)(p^2+q^2+r^2+s^2) \\ && L_1^2 + L_2^2+L_3^2+L_4^2 &= \ldots \end{align*}