\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i5) M:matrix([a11,a12,a13],[a12,a22,a23],[a13,a23,a33]);
Xr:matrix([x,y,1]);
Xc:matrix([x],[y],[1]);
\[\tag{M}\begin{pmatrix}\mathit{a11} & \mathit{a12} & \mathit{a13}\\ \mathit{a12} & \mathit{a22} & \mathit{a23}\\ \mathit{a13} & \mathit{a23} & \mathit{a33}\end{pmatrix}\] \[\tag{Xr}\begin{pmatrix}x & y & 1\end{pmatrix}\] \[\tag{Xc}\begin{pmatrix}x\\ y\\ 1\end{pmatrix}\]
(%i17) K:expand(Xr.M.Xc);
\[\tag{K}\mathit{a22}\, {{y}^{2}}+2 \mathit{a12} x y+2 \mathit{a23} y+\mathit{a11}\, {{x}^{2}}+2 \mathit{a13} x+\mathit{a33}\]
(%i18) K1:expand(subst([x=x1·cos(%alpha)−y1·sin(%alpha), y=x1·sin(%alpha)+y1·cos(%alpha)],K));
\[\tag{K1}{{\cos{\left( \alpha \right) }}^{2}}\, \mathit{a22}\, {{\mathit{y1}}^{2}}-2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a12}\, {{\mathit{y1}}^{2}}+{{\sin{\left( \alpha \right) }}^{2}}\, \mathit{a11}\, {{\mathit{y1}}^{2}}+2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a22}\, \mathit{x1}\, \mathit{y1}-2 {{\sin{\left( \alpha \right) }}^{2}}\, \mathit{a12}\, \mathit{x1}\, \mathit{y1}+2 {{\cos{\left( \alpha \right) }}^{2}}\, \mathit{a12}\, \mathit{x1}\, \mathit{y1}-2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a11}\, \mathit{x1}\, \mathit{y1}+2 \cos{\left( \alpha \right) } \mathit{a23}\, \mathit{y1}-2 \sin{\left( \alpha \right) } \mathit{a13}\, \mathit{y1}+{{\sin{\left( \alpha \right) }}^{2}}\, \mathit{a22}\, {{\mathit{x1}}^{2}}+2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a12}\, {{\mathit{x1}}^{2}}+{{\cos{\left( \alpha \right) }}^{2}}\, \mathit{a11}\, {{\mathit{x1}}^{2}}+2 \sin{\left( \alpha \right) } \mathit{a23}\, \mathit{x1}+2 \cos{\left( \alpha \right) } \mathit{a13}\, \mathit{x1}+\mathit{a33}\]
(%i38) K1xx:coeff(K1,x1,2);
\[\tag{K1xx}{{\sin{\left( \alpha \right) }}^{2}}\, \mathit{a22}+2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a12}+{{\cos{\left( \alpha \right) }}^{2}}\, \mathit{a11}\]
(%i36) K1xy:coeff(coeff(K1,x1),y1);
\[\tag{K1xy}2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a22}-2 {{\sin{\left( \alpha \right) }}^{2}}\, \mathit{a12}+2 {{\cos{\left( \alpha \right) }}^{2}}\, \mathit{a12}-2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a11}\]
(%i39) K1yy:coeff(K1,y1,2);
\[\tag{K1yy}{{\cos{\left( \alpha \right) }}^{2}}\, \mathit{a22}-2 \cos{\left( \alpha \right) } \sin{\left( \alpha \right) } \mathit{a12}+{{\sin{\left( \alpha \right) }}^{2}}\, \mathit{a11}\]
(%i46) K1x:coeff(coeff(K1,x1,1),y1,0);
\[\tag{K1x}2 \sin{\left( \alpha \right) } \mathit{a23}+2 \cos{\left( \alpha \right) } \mathit{a13}\]
(%i47) K1y:coeff(coeff(K1,y1,1),x1,0);
\[\tag{K1y}2 \cos{\left( \alpha \right) } \mathit{a23}-2 \sin{\left( \alpha \right) } \mathit{a13}\]
(%i51) factor(trigreduce(K1xy));
\[\tag{%o51} \sin{\left( 2 \alpha \right) } \mathit{a22}+2 \cos{\left( 2 \alpha \right) } \mathit{a12}-\sin{\left( 2 \alpha \right) } \mathit{a11}\]
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