Math, asked by Balajinaik4544, 11 hours ago

10. Reduce the Quadratic form2x2 + 2y2 + 2z2 - 2xy + 2xz - 2yz into the canonical form byOrthogonal transformation and discuss its nature.​

Answers

Answered by Anonymous
5

We know that signature is the difference between the number of positive and negative square terms of a quadratic form. Therefore, we have signature = 2 – 1 = 1

Answered by ck4958653
0

Answer:

The given quadratic form is

3x^2+5y^2+3z^2-2xy-2yz+2xz3x

2

+5y

2

+3z

2

−2xy−2yz+2xz

The matrix of the given quadratic form is

A=\begin{pmatrix} 3 & -1 &1 \\ -1& 5 & -1 \\ 1 & -1 & 3 \end{pmatrix}A=

3

−1

1

−1

5

−1

1

−1

3

We write , A= IAIA=IAI

i,e, \begin {pmatrix} 3&-1&1 \\ -1&5&-1 \\ 1&-1&3 \end{pmatrix}

3

−1

1

−1

5

−1

1

−1

3

=\begin{pmatrix} 1&0&0 \\ 0&1&0\\ 0&0&1 \end{pmatrix}=

1

0

0

0

1

0

0

0

1

A\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{pmatrix}A

1

0

0

0

1

0

0

0

1

Now we shall reduce AA 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,R

2

→3R

2

+R

1

,R

3

→3R

3

−R

1

;C

2

→C

2

+

3

1

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_2C

3

→C

3

3

1

C

1

;R

3

→7R

3

+R

2

;C

3

→C

3

7

1

C

2

We get ,

\begin{pmatrix} 3&0&0\\ 0&14&0\\ 0&0&54 \end{pmatrix}

3

0

0

0

14

0

0

0

54

== \begin{pmatrix} 1&0&0 \\ 1&3&0\\ -6&3&21 \end{pmatrix}

1

1

−6

0

3

3

0

0

21

A\begin{pmatrix} 1& \frac{1}{3} & \frac{-8}{21} \\ 0&1&\frac{-1}{7} \\ 0&0&1 \end{pmatrix}A

1

0

0

3

1

1

0

21

−8

7

−1

1

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