This article answers the first part of problem 2 in Mel Nathanson's article "Matrix Scaling, Explicit Sinkhorn limits, and Arithmetic" --------------------------------------------------------------------------------------------------------- The exact value of the Sinkhorn limit for a certain symmetric matrix of size, 3 By Shalsoh B. Ekhad Consider the following symmetric matrix [K 1 1] [ ] [1 L 1] [ ] [1 1 1] Let , z, be the smallest real positive root of the polynomial equation 2 2 L + (-4 K L + 1) z + (6 K L - 2 K L - 3 K - 1) z 2 3 2 4 - (K - 1) (4 K L - 3 K - 1) z + K (K - 1) (K L - 1) z = 0 and in Maple notation L+(-4*K*L+1)*z+(6*K^2*L-2*K*L-3*K-1)*z^2-(K-1)*(4*K^2*L-3*K-1)*z^3+K*(K-1)^2*(K*L-1)*z^4 = 0 and in LaTex L+ \left( -4\,LK+1 \right) z+ \left( 6\,L{K}^{2}-2\,LK-3\,K-1 \right) { z}^{2}- \left( K-1 \right) \left( 4\,L{K}^{2}-3\,K-1 \right) {z}^{3}+K \left( K-1 \right) ^{2} \left( LK-1 \right) {z}^{4}=0 NULL Then the diagonal matrix X such that S=XMX is doubly stochastic has in its d\ iagonal / 4 2 4 5 3 2 3 \ 1/2 | 3 L K 3 K K L K K 3 K K L 3 K L| 3 z [1, |- ------ - ----- - ----- + ----- + ----- + ----- - ----- + ------| z \ L - 1 L - 1 L - 1 L - 1 L - 1 L - 1 L - 1 L - 1 / / 3 2 4 2 3 \ | 1 2 K 3 K L 3 L K 3 K 6 K L| 2 + |----- + ----- - ------ - ------ - ----- + ------| z \L - 1 L - 1 L - 1 L - 1 L - 1 L - 1 / / 2 3 2 \ 2 / | K 2 K L 3 K L K 5 K L| 1 2 K L K L | + |- ----- + ----- + ------ + ----- - ------| z - ----- + ----- - -----, | \ L - 1 L - 1 L - 1 L - 1 L - 1 / L - 1 L - 1 L - 1 \ 3 3 2 4 4 2 K 4 K K 2 K 2 K 2 L K 2 K - --------- - ----- - --------- + --------- + ----- + ------ + ----- L (L - 1) L - 1 L (L - 1) L (L - 1) L - 1 L - 1 L - 1 3 5 \ / 2 2 4 3 2 K L L K | 3 | K L 3 K 3 L K 6 K 4 K - ----- - -----| z + |- ----- + --------- + ------ - ----- + ----- L - 1 L - 1/ \ L - 1 L (L - 1) L - 1 L - 1 L - 1 3 \ 2 K L 2 K 1 2 K | 2 - ------ - --------- - --------- + -----| z L - 1 L (L - 1) L (L - 1) L - 1/ / 2 3 2 \ 2 |7 K 1 3 K L K L 1 3 K | K L + |----- + ----- - ------ - ----- - --------- - ---------| z + ----- \L - 1 L - 1 L - 1 L - 1 L (L - 1) L (L - 1)/ L - 1 1 K L 3 K + --------- + ----- - -----] L (L - 1) L - 1 L - 1 and in Maple notation z^(1/2)*[1, (-3*L/(L-1)*K^4-3/(L-1)*K^2-1/(L-1)*K^4+L/(L-1)*K^5+1/(L-1)*K+3/(L-\ 1)*K^3-1/(L-1)*K^2*L+3/(L-1)*K^3*L)*z^3+(1/(L-1)+2/(L-1)*K^3-3/(L-1)*K^2*L-3*L/ (L-1)*K^4-3/(L-1)*K^2+6/(L-1)*K^3*L)*z^2+(-1/(L-1)*K^2+2/(L-1)*K*L+3/(L-1)*K^3* L+1/(L-1)*K-5/(L-1)*K^2*L)*z-1/(L-1)+2/(L-1)*K*L-1/(L-1)*K^2*L, (-1/L/(L-1)*K^3 -4/(L-1)*K^3-1/L/(L-1)*K+2/L/(L-1)*K^2+2/(L-1)*K^4+2*L/(L-1)*K^4+2/(L-1)*K^2-1/ (L-1)*K^3*L-L/(L-1)*K^5)*z^3+(-1/(L-1)*K^2*L+3/L/(L-1)*K^2+3*L/(L-1)*K^4-6/(L-1 )*K^3+4/(L-1)*K^2-2/(L-1)*K^3*L-2/L/(L-1)*K-1/L/(L-1)+2/(L-1)*K)*z^2+(7/(L-1)*K ^2+1/(L-1)-3/(L-1)*K^3*L-1/(L-1)*K^2*L-1/L/(L-1)-3/L/(L-1)*K)*z+1/(L-1)*K^2*L+1 /L/(L-1)+1/(L-1)*K*L-3/(L-1)*K] and the Sinkhorn limit is 3 2 2 3 (K L - 2 K L - K + K L + 2 K - 1) K z (3 K L - L - 2) K z [[K z, - ----------------------------------------- - ------------------- L - 1 L - 1 3 2 2 2 (3 K L - 3 K L - 2 K + K + 1) z K L - 1 + ----------------------------------- + -------, L - 1 L - 1 3 2 2 3 (K L - 2 K L - K + K L + 2 K - 1) K z (3 K L - 2 L - 1) K z ----------------------------------------- + --------------------- L - 1 L - 1 3 2 2 2 (3 K L - 3 K L - 2 K + K + 1) z (K - 1) L - ----------------------------------- - ---------], [ L - 1 L - 1 3 2 2 3 (K L - 2 K L - K + K L + 2 K - 1) K z (3 K L - L - 2) K z - ----------------------------------------- - ------------------- L - 1 L - 1 3 2 2 2 (3 K L - 3 K L - 2 K + K + 1) z K L - 1 + ----------------------------------- + -------, L - 1 L - 1 3 2 2 3 (K L - 2 K L - K + K L + 2 K - 1) K z ----------------------------------------- L - 1 3 2 2 2 2 (4 K L - 4 K L - 3 K + 2 K + 1) z (5 K L - K L - 3 K - 1) z - ------------------------------------- + -------------------------- L - 1 L - 1 2 2 2 2 K L - L - 1 (K L - K L - K + 1) K z (2 K L - K - 1) z K L - 1 - -------------, ------------------------- - ------------------ + -------] L - 1 L - 1 L - 1 L - 1 3 2 2 3 (K L - 2 K L - K + K L + 2 K - 1) K z (3 K L - 2 L - 1) K z , [----------------------------------------- + --------------------- L - 1 L - 1 3 2 2 2 (3 K L - 3 K L - 2 K + K + 1) z (K - 1) L - ----------------------------------- - ---------, L - 1 L - 1 2 2 2 (K L - K L - K + 1) K z (2 K L - K - 1) z K L - 1 ------------------------- - ------------------ + -------, L - 1 L - 1 L - 1 3 2 2 3 3 2 2 2 (K L - 2 K L - K + K L + 2 K - 1) K z (2 K L - 2 K L - K + 1) z - ----------------------------------------- + ----------------------------- L - 1 L - 1 2 (K L - 2 K L + 1) z - --------------------]] L - 1 and in Maple notation [[K*z, -(K^3*L-2*K^2*L-K^2+K*L+2*K-1)/(L-1)*K*z^3-(3*K*L-L-2)/(L-1)*K*z+(3*K^3* L-3*K^2*L-2*K^2+K+1)/(L-1)*z^2+(K*L-1)/(L-1), (K^3*L-2*K^2*L-K^2+K*L+2*K-1)/(L-\ 1)*K*z^3+(3*K*L-2*L-1)/(L-1)*K*z-(3*K^3*L-3*K^2*L-2*K^2+K+1)/(L-1)*z^2-(K-1)/(L -1)*L], [-(K^3*L-2*K^2*L-K^2+K*L+2*K-1)/(L-1)*K*z^3-(3*K*L-L-2)/(L-1)*K*z+(3*K^ 3*L-3*K^2*L-2*K^2+K+1)/(L-1)*z^2+(K*L-1)/(L-1), (K^3*L-2*K^2*L-K^2+K*L+2*K-1)/( L-1)*K*z^3-(4*K^3*L-4*K^2*L-3*K^2+2*K+1)/(L-1)*z^2+(5*K^2*L-K*L-3*K-1)/(L-1)*z- (2*K*L-L-1)/(L-1), (K^2*L-K*L-K+1)/(L-1)*K*z^2-(2*K^2*L-K-1)/(L-1)*z+(K*L-1)/(L -1)], [(K^3*L-2*K^2*L-K^2+K*L+2*K-1)/(L-1)*K*z^3+(3*K*L-2*L-1)/(L-1)*K*z-(3*K^3 *L-3*K^2*L-2*K^2+K+1)/(L-1)*z^2-(K-1)/(L-1)*L, (K^2*L-K*L-K+1)/(L-1)*K*z^2-(2*K ^2*L-K-1)/(L-1)*z+(K*L-1)/(L-1), -(K^3*L-2*K^2*L-K^2+K*L+2*K-1)/(L-1)*K*z^3+(2* K^3*L-2*K^2*L-K^2+1)/(L-1)*z^2-(K^2*L-2*K*L+1)/(L-1)*z]]