Answer to Question #108260 in Linear Algebra for Kentse

Question #108260

Find the entries xi;i D 1; 2; 3; 4 in the matrix P so that P is an orthogonal matrix where
P D
1
2
2
6
6
4
1 1 1 1
1 1

Expert's answer

Let matrix "P" be

"P=k\\begin{pmatrix}\n 1 & a & b & c \\\\\n d & e & x_1 & x_2 \\\\\n 1 & 1 & x_3 & x_4 \\\\\n 1 & 1 & 1 & 1\n\\end{pmatrix}"

Let

"r_1=\\begin{pmatrix}\n 1 \\\\\n d \\\\\n 1 \\\\\n 1\n\\end{pmatrix},r_2=\\begin{pmatrix}\n a \\\\\n e \\\\\n 1 \\\\\n 1\n\\end{pmatrix},r_3=\\begin{pmatrix}\n b \\\\\n x_1 \\\\\n x_3 \\\\\n 1\n\\end{pmatrix}, r_4=\\begin{pmatrix}\n c \\\\\n x_2 \\\\\n x_4 \\\\\n 1\n\\end{pmatrix}"

"P" is an orthogonal matrix

"PP^T=I"

Then

"r_1r_2=r_1r_3=r_1r_4=r_2r_3=r_2r_4=r_3r_4=0""a+ed+1+1=0""b+dx_1+x_3+1=0""c+dx_2+x_4+1=0""ab+ex_1+x_3+1=0""ac+ex_2+x_4+1=0""bc+x_1x_2+x_3x_4+1=0"

Let "d=1." Then

"a+e+2=0""b+x_1+x_3+1=0""c+x_2+x_4+1=0""ab+ex_1+x_3+1=0""ac+ex_2+x_4+1=0""bc+x_1x_2+x_3x_4+1=0"

Solve

"a=-1, b=-1,c=1, d=1, e=-1,"

"x_1=1, x_2=-1,x_3=-1, x_4=-1"

"k^2r_1^2=k^2r_2^2=k^2r_3^2=k^2r_4^2=1"

"k=\\pm0.5"

"P=0.5\\begin{pmatrix}\n 1 & -1 & -1 & 1 \\\\\n 1 & -1 & 1 & -1 \\\\\n 1 & 1 & -1 & -1 \\\\\n 1 & 1 & 1 & 1\n\\end{pmatrix},P^T=0.5\\begin{pmatrix}\n 1 & 1 & 1 & 1 \\\\\n -1 & -1 & 1 & 1 \\\\\n -1 & 1 & -1 & 1 \\\\\n 1 & 1 & -1 & 1\n\\end{pmatrix}"

"PP^T=0.5\\begin{pmatrix}\n 1 & -1 & -1 & 1 \\\\\n 1 & -1 & 1 & -1 \\\\\n 1 & 1 & -1 & -1 \\\\\n 1 & 1 & 1 & 1\n\\end{pmatrix}\\begin{pmatrix}\n 1 & 1 & 1 & 1 \\\\\n -1 & -1 & 1 & 1 \\\\\n -1 & 1 & -1 & 1 \\\\\n 1 & -1 & -1 & 1\n\\end{pmatrix}="

"=\\begin{pmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n\\end{pmatrix}=I"

"x_1=1, x_2=-1,x_3=-1, x_4=-1"