线性代数f=x1^2+2x2^2+5x3^2+2x1x2+2x1x3+6x2x3=y1^2+y2^2,和解？

Question

如题，线性代数f=x1^2+2x2^2+5x3^2+2x1x2+2x1x3+6x2x3=y1^2+y2^2
求解为何等于y1^2+y2^2

Answer 1

直接配方即可
f=x1^2+2x2^2+5x3^2+2x1x2+2x1x3+6x2x3

=x1^2+x2^2+x3^2+2x1x2+2x1x3+2x2x3+x2^2+4x3^2+4x2x3

=(x1+x2+x3)^2+(x2+2x3)^2
设 y1=x1+x2+x3 y2=x2+2x3
则 f=x1^2+2x2^2+5x3^2+2x1x2+2x1x3+6x2x3=y1^2+y2^2