Question

classify the quadratic forms {x {1} }^{2 } + {x {3} }^{2}x1 2 +x3 2​

Answer 1

Step-by-step explanation:





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Answer to Question #153072 in Linear Algebra for Kishore Kris

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Linear Algebra

Question #153072

3. Reduce the quadratic form x 1 ^ 2 +5 x 2 ^ 2 + x 3 ^ 2 +2x 1 x 2 +2x 2 x 3 +6x 3 x 1 to canonical form through an orthogonal transformation and also find its rank, index, signature and nature of the quadratic form.

Expert's answer

The matrix of the quadratic form is

A=\begin{bmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \\ \end{bmatrix}A=⎣⎢⎡113151311⎦⎥⎤

Find the eigenvalues and eigenvectors of A

A-\lambda I=\begin{bmatrix} 1-\lambda & 1 & 3 \\ 1 & 5-\lambda & 1 \\ 3 & 1 & 1-\lambda \\ \end{bmatrix}A−λI=⎣⎢⎡1−λ1315−λ1311−λ⎦⎥⎤

\begin{vmatrix} 1-\lambda & 1 & 3 \\ 1 & 5-\lambda & 1 \\ 3 & 1 & 1-\lambda \\ \end{vmatrix}=(1-\lambda)\begin{vmatrix} 5-\lambda & 1 \\ 1 & 1-\lambda \end{vmatrix}∣∣∣∣∣∣∣1−λ1315−λ1311−λ∣∣∣∣∣∣∣=(1−λ)∣∣∣∣∣5−λ111−λ∣∣∣∣∣

-(1)\begin{vmatrix} 1 & 1 \\ 3 & 1-\lambda \end{vmatrix}+(3)\begin{vmatrix} 1 & 5-\lambda \\ 3 & 1 \end{vmatrix}−(1)∣∣∣∣∣1311−λ∣∣∣∣∣+(3)∣∣∣∣∣135−λ1∣∣∣∣∣

=(1-\lambda)^2(5-\lambda)-1+\lambda-1+\lambda+3+3-45+9\lambda=(1−λ)2(5−λ)−1+λ−1+λ+3+3−45+9λ

=-\lambda^3+7\lambda^2-36=−λ3+7λ2−36

This is a characteristic polynomial.

Solve the equation

-\lambda^3+7\lambda^2-36=0−λ3+7λ2−36=0

-\lambda^2(\lambda-6)+(\lambda-6)(\lambda+6)=0−λ2(λ−6)+(λ−6)(λ+6)=0

-(\lambda-6)(\lambda-3)(\lambda+2)=0−(λ−6)(λ