Answer to Question #248558 in Linear Algebra for chaitu

Question #248558

26. Reduce the Quadratic Form (Q.F.) 2x

2 + 5y

2 + 3z

2 + 4xy to canonical form by an

orthogonal transformation. Also find its nature, rank, index and signature of the Q.F.


1
Expert's answer
2021-10-12T09:01:05-0400

The given quadratic form is

"3x^2+5y^2+3z^2-2xy-2yz+2xz"

The matrix of the given quadratic form is

"A=\\begin{pmatrix}\n3 & -1 &1 \\\\\n-1& 5 & -1 \\\\\n1 & -1 & 3 \n\\end{pmatrix}"


We write , "A= IAI"

i,e, "\\begin {pmatrix}\n3&-1&1 \\\\\n-1&5&-1 \\\\\n1&-1&3\n\\end{pmatrix}" "=\\begin{pmatrix}\n1&0&0 \\\\\n0&1&0\\\\\n0&0&1\n\\end{pmatrix}" "A\\begin{pmatrix}\n1&0&0 \\\\\n0&1&0 \\\\\n0&0&1\n\\end{pmatrix}"

Now we shall reduce "A" to diagonal form by applying congruence operation on it . Performing "R_2\\rightarrow 3R_2+R_1,R_3\\rightarrow 3R_3-R_1;C_2\\rightarrow C_2+\\frac{1}{3}C_1,"


"C_3\\rightarrow C_3-\\frac{1}{3}C_1;R_3\\rightarrow 7R_3+R_2;C_3\\rightarrow C_3-\\frac{1}{7}C_2"

We get ,

"\\begin{pmatrix}\n3&0&0\\\\\n0&14&0\\\\\n0&0&54\n\\end{pmatrix}" "=" "\\begin{pmatrix}\n1&0&0 \\\\\n1&3&0\\\\\n-6&3&21\n\\end{pmatrix}" "A\\begin{pmatrix}\n1& \\frac{1}{3} & \\frac{-8}{21} \\\\\n0&1&\\frac{-1}{7} \\\\\n0&0&1 \n\\end{pmatrix}"


Performing , "R_1\\rightarrow \\frac{1}{\\sqrt{3}}R_1,\nC_1\\rightarrow \\frac{1}{\\sqrt{3}}C_1" ;

"R_2\\rightarrow \\frac{1}{\\sqrt{14}}R_1,C_2\\rightarrow \\frac{1} {\\sqrt{14}};"

"R_3\\rightarrow \\frac{1}{\\sqrt{54}}R_3,C_3\\rightarrow \\frac{1}{\\sqrt{54}}C_2"

We get ,

"\\begin{pmatrix}\n1&0&0 \\\\\n0&1&0\\\\\n0&0&1\n\\end{pmatrix}" "=\\begin{pmatrix}\na&0&0\\\\\nb&3b&0\\\\\n-6c&3c&21c\n\\end{pmatrix}A" "\\begin{pmatrix}\na&\\frac{b}{3}&\\frac{-8c}{21} \\\\\n0&b&\\frac{-c}{7} \\\\\n0&0&c\n\\end{pmatrix}"

Where "a=\\frac{1}{\\sqrt{3}},b=\\frac{1}{\\sqrt{14}},c=\\frac{1}{\\sqrt{54}}"

Thus the linear transformation ,

"X=PY" where ,


"P=\\begin{pmatrix}\na&\\frac{b}{3}&\\frac{-8c}{21} \\\\\n0&b&\\frac{-c}{7} \\\\\n0&0&c\n\\end{pmatrix}"


"X=[x,y,z]',Y=[y_1,y_2,y_3]'"

This transformation gives quadratic form to the normal form

"{y_1}^2+{y_2}^2+{y_3}^2.............(1)"

The rank "r" of the given quadratic form "=" The number of nonzero terms in its normal form "(1)"

"=" 3.

The signature of the given quadratic form "=" the excess of the number of positive terms over the number of negative terms in its normal form"=3-0=3"

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