Let matrix P be
P=k⎝⎛1d11ae11bx1x31cx2x41⎠⎞ Let
r1=⎝⎛1d11⎠⎞,r2=⎝⎛ae11⎠⎞,r3=⎝⎛bx1x31⎠⎞,r4=⎝⎛cx2x41⎠⎞ P is an orthogonal matrix
PPT=I Then
r1r2=r1r3=r1r4=r2r3=r2r4=r3r4=0a+ed+1+1=0b+dx1+x3+1=0c+dx2+x4+1=0ab+ex1+x3+1=0ac+ex2+x4+1=0bc+x1x2+x3x4+1=0 Let d=1. Then
a+e+2=0b+x1+x3+1=0c+x2+x4+1=0ab+ex1+x3+1=0ac+ex2+x4+1=0bc+x1x2+x3x4+1=0 Solve
a=−1,b=−1,c=1,d=1,e=−1,
x1=1,x2=−1,x3=−1,x4=−1
k2r12=k2r22=k2r32=k2r42=1
k=±0.5
P=0.5⎝⎛1111−1−111−11−111−1−11⎠⎞,PT=0.5⎝⎛1−1−111−11111−1−11111⎠⎞
PPT=0.5⎝⎛1111−1−111−11−111−1−11⎠⎞⎝⎛1−1−111−11−111−1−11111⎠⎞=
=⎝⎛1000010000100001⎠⎞=I
x1=1,x2=−1,x3=−1,x4=−1
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