Nonlinear system identification

In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form

$\begin{array}{rl}& (x_1^2+x_2^2+x_3^2+x_4^2+\cdots +x_{16}^2)(y_1^2+y_2^2+y_3^2+y_4^2+\cdots +y_{16}^2)\\ & =z_1^2+z_2^2+z_3^2+z_4^2+\cdots +z_{16}^2\end{array}$

It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Pfister around the same time. There are several versions, a concise one of which is

${}^{z_1=x_1{y}_1-x_2{y}_2-x_3{y}_3-x_4{y}_4-x_5{y}_5-x_6{y}_6-x_7{y}_7-x_8{y}_8+u_1{y}_9-u_2{y}_{10}-u_3{y}_{11}-u_4{y}_{12}-u_5{y}_{13}-u_6{y}_{14}-u_7{y}_{15}-u_8{y}_{16}}$

${}^{z_2=x_2{y}_1+x_1{y}_2+x_4{y}_3-x_3{y}_4+x_6{y}_5-x_5{y}_6-x_8{y}_7+x_7{y}_8+u_2{y}_9+u_1{y}_{10}+u_4{y}_{11}-u_3{y}_{12}+u_6{y}_{13}-u_5{y}_{14}-u_8{y}_{15}+u_7{y}_{16}}$

${}^{z_3=x_3{y}_1-x_4{y}_2+x_1{y}_3+x_2{y}_4+x_7{y}_5+x_8{y}_6-x_5{y}_7-x_6{y}_8+u_3{y}_9-u_4{y}_{10}+u_1{y}_{11}+u_2{y}_{12}+u_7{y}_{13}+u_8{y}_{14}-u_5{y}_{15}-u_6{y}_{16}}$

${}^{z_4=x_4{y}_1+x_3{y}_2-x_2{y}_3+x_1{y}_4+x_8{y}_5-x_7{y}_6+x_6{y}_7-x_5{y}_8+u_4{y}_9+u_3{y}_{10}-u_2{y}_{11}+u_1{y}_{12}+u_8{y}_{13}-u_7{y}_{14}+u_6{y}_{15}-u_5{y}_{16}}$

${}^{z_5=x_5{y}_1-x_6{y}_2-x_7{y}_3-x_8{y}_4+x_1{y}_5+x_2{y}_6+x_3{y}_7+x_4{y}_8+u_5{y}_9-u_6{y}_{10}-u_7{y}_{11}-u_8{y}_{12}+u_1{y}_{13}+u_2{y}_{14}+u_3{y}_{15}+u_4{y}_{16}}$

${}^{z_6=x_6{y}_1+x_5{y}_2-x_8{y}_3+x_7{y}_4-x_2{y}_5+x_1{y}_6-x_4{y}_7+x_3{y}_8+u_6{y}_9+u_5{y}_{10}-u_8{y}_{11}+u_7{y}_{12}-u_2{y}_{13}+u_1{y}_{14}-u_4{y}_{15}+u_3{y}_{16}}$

${}^{z_7=x_7{y}_1+x_8{y}_2+x_5{y}_3-x_6{y}_4-x_3{y}_5+x_4{y}_6+x_1{y}_7-x_2{y}_8+u_7{y}_9+u_8{y}_{10}+u_5{y}_{11}-u_6{y}_{12}-u_3{y}_{13}+u_4{y}_{14}+u_1{y}_{15}-u_2{y}_{16}}$

${}^{z_8=x_8{y}_1-x_7{y}_2+x_6{y}_3+x_5{y}_4-x_4{y}_5-x_3{y}_6+x_2{y}_7+x_1{y}_8+u_8{y}_9-u_7{y}_{10}+u_6{y}_{11}+u_5{y}_{12}-u_4{y}_{13}-u_3{y}_{14}+u_2{y}_{15}+u_1{y}_{16}}$

${}^{z_9=x_9{y}_1-x_{10}{y}_2-x_{11}{y}_3-x_{12}{y}_4-x_{13}{y}_5-x_{14}{y}_6-x_{15}{y}_7-x_{16}{y}_8+x_1{y}_9-x_2{y}_{10}-x_3{y}_{11}-x_4{y}_{12}-x_5{y}_{13}-x_6{y}_{14}-x_7{y}_{15}-x_8{y}_{16}}$

${}^{z_{10}=x_{10}{y}_1+x_9{y}_2+x_{12}{y}_3-x_{11}{y}_4+x_{14}{y}_5-x_{13}{y}_6-x_{16}{y}_7+x_{15}{y}_8+x_2{y}_9+x_1{y}_{10}+x_4{y}_{11}-x_3{y}_{12}+x_6{y}_{13}-x_5{y}_{14}-x_8{y}_{15}+x_7{y}_{16}}$

${}^{z_{11}=x_{11}{y}_1-x_{12}{y}_2+x_9{y}_3+x_{10}{y}_4+x_{15}{y}_5+x_{16}{y}_6-x_{13}{y}_7-x_{14}{y}_8+x_3{y}_9-x_4{y}_{10}+x_1{y}_{11}+x_2{y}_{12}+x_7{y}_{13}+x_8{y}_{14}-x_5{y}_{15}-x_6{y}_{16}}$

${}^{z_{12}=x_{12}{y}_1+x_{11}{y}_2-x_{10}{y}_3+x_9{y}_4+x_{16}{y}_5-x_{15}{y}_6+x_{14}{y}_7-x_{13}{y}_8+x_4{y}_9+x_3{y}_{10}-x_2{y}_{11}+x_1{y}_{12}+x_8{y}_{13}-x_7{y}_{14}+x_6{y}_{15}-x_5{y}_{16}}$

${}^{z_{13}=x_{13}{y}_1-x_{14}{y}_2-x_{15}{y}_3-x_{16}{y}_4+x_9{y}_5+x_{10}{y}_6+x_{11}{y}_7+x_{12}{y}_8+x_5{y}_9-x_6{y}_{10}-x_7{y}_{11}-x_8{y}_{12}+x_1{y}_{13}+x_2{y}_{14}+x_3{y}_{15}+x_4{y}_{16}}$

${}^{z_{14}=x_{14}{y}_1+x_{13}{y}_2-x_{16}{y}_3+x_{15}{y}_4-x_{10}{y}_5+x_9{y}_6-x_{12}{y}_7+x_{11}{y}_8+x_6{y}_9+x_5{y}_{10}-x_8{y}_{11}+x_7{y}_{12}-x_2{y}_{13}+x_1{y}_{14}-x_4{y}_{15}+x_3{y}_{16}}$

${}^{z_{15}=x_{15}{y}_1+x_{16}{y}_2+x_{13}{y}_3-x_{14}{y}_4-x_{11}{y}_5+x_{12}{y}_6+x_9{y}_7-x_{10}{y}_8+x_7{y}_9+x_8{y}_{10}+x_5{y}_{11}-x_6{y}_{12}-x_3{y}_{13}+x_4{y}_{14}+x_1{y}_{15}-x_2{y}_{16}}$

${}^{z_{16}=x_{16}{y}_1-x_{15}{y}_2+x_{14}{y}_3+x_{13}{y}_4-x_{12}{y}_5-x_{11}{y}_6+x_{10}{y}_7+x_9{y}_8+x_8{y}_9-x_7{y}_{10}+x_6{y}_{11}+x_5{y}_{12}-x_4{y}_{13}-x_3{y}_{14}+x_2{y}_{15}+x_1{y}_{16}}$

where the $u_i$ are,

$u_1=\frac{(a{x}_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_9-2x_1(b{x}_1{x}_9+x_2{x}_{10}+x_3{x}_{11}+x_4{x}_{12}+x_5{x}_{13}+x_6{x}_{14}+x_7{x}_{15}+x_8{x}_{16})}{c}$

$u_2=\frac{(x_1^2+a{x}_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{10}-2x_2(x_1{x}_9+b{x}_2{x}_{10}+x_3{x}_{11}+x_4{x}_{12}+x_5{x}_{13}+x_6{x}_{14}+x_7{x}_{15}+x_8{x}_{16})}{c}$

$u_3=\frac{(x_1^2+x_2^2+a{x}_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{11}-2x_3(x_1{x}_9+x_2{x}_{10}+b{x}_3{x}_{11}+x_4{x}_{12}+x_5{x}_{13}+x_6{x}_{14}+x_7{x}_{15}+x_8{x}_{16})}{c}$

$u_4=\frac{(x_1^2+x_2^2+x_3^2+a{x}_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{12}-2x_4(x_1{x}_9+x_2{x}_{10}+x_3{x}_{11}+b{x}_4{x}_{12}+x_5{x}_{13}+x_6{x}_{14}+x_7{x}_{15}+x_8{x}_{16})}{c}$

$u_5=\frac{(x_1^2+x_2^2+x_3^2+x_4^2+a{x}_5^2+x_6^2+x_7^2+x_8^2)x_{13}-2x_5(x_1{x}_9+x_2{x}_{10}+x_3{x}_{11}+x_4{x}_{12}+b{x}_5{x}_{13}+x_6{x}_{14}+x_7{x}_{15}+x_8{x}_{16})}{c}$

$u_6=\frac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+a{x}_6^2+x_7^2+x_8^2)x_{14}-2x_6(x_1{x}_9+x_2{x}_{10}+x_3{x}_{11}+x_4{x}_{12}+x_5{x}_{13}+b{x}_6{x}_{14}+x_7{x}_{15}+x_8{x}_{16})}{c}$

$u_7=\frac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+a{x}_7^2+x_8^2)x_{15}-2x_7(x_1{x}_9+x_2{x}_{10}+x_3{x}_{11}+x_4{x}_{12}+x_5{x}_{13}+x_6{x}_{14}+b{x}_7{x}_{15}+x_8{x}_{16})}{c}$

$u_8=\frac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+a{x}_8^2)x_{16}-2x_8(x_1{x}_9+x_2{x}_{10}+x_3{x}_{11}+x_4{x}_{12}+x_5{x}_{13}+x_6{x}_{14}+x_7{x}_{15}+b{x}_8{x}_{16})}{c}$

and,

$a=-1,b=0,c=x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2.$

The $u_i$ also obey,

$u_1^2+u_2^2+u_3^2+u_4^2+u_5^2+u_6^2+u_7^2+u_8^2=x_9^2+x_{10}^2+x_{11}^2+x_{12}^2+x_{13}^2+x_{14}^2+x_{15}^2+x_{16}^2$

Thus the identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen rational squares. If all $x_i,y_i$ with $i>8$ are set equal to zero, then it reduces to the Degen's eight-square.

No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form

$(x_1^2+x_2^2+x_3^2+\cdots +x_n^2)(y_1^2+y_2^2+y_3^2+\cdots +y_n^2)=z_1^2+z_2^2+z_3^2+\cdots +z_n^2$

with the $z_i$ bilinear functions of the $x_i$ and $y_i$ is possible only for n ∈ {1, 2, 4, 8} . However, the more general Pfister's theorem (1965) shows that if the $z_i$ are just rational functions of one set of variables, hence has a denominator, then it is possible for all $n=2^m$. There are also non-bilinear versions of Euler's four-square and Degen's eight-square identities.

See also

• Brahmagupta–Fibonacci identity
• Euler's four-square identity
• Degen's eight-square identity
• Sedenions

External links

• Pfister's 16-Square Identity