求该二次型的正交变换:2x1^2+x2^2-4x1x2-4x2x3

2025-05-20 00:49:30
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回答(1):

配方法:

2x1²+x2²-4x1x2-4x2x3
=2x1²-4x1x2+2x2²-x2²-4x2x3-4x3²+4x3²
=2(x1-x2)²-(x2+x3)²+4x3²
设y1=√2(x1-x2),y2=x2+x3,y3=2x3;
x3=y3/2,x2=y2-x3=y2-y3/2,x1=y1/√2+x2=y1/√2+y2-y3/2
标准型:y1²-y2²+y3²