Quadratic forms

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Problem: Investigate the quadratic form for signdefiniteness, bring it to the canonical form by orthogonal transformation and write down the transition matrix. \[ \begin{array}{l} f\left(x_{1}, x_{2} x_{3}\right)=3 x_{1}{ }^{2}+x_{2}{ }^{2}-\frac{3}{2} x_{3}{ }^{2}+2 \sqrt{3} x_{1} x_{2}- \\ -x_{1} x_{3}+\sqrt{3} x_{2} x_{3} . \end{array} \]

1.8.1 Quadratic forms

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Problem: At what values of \( \lambda \) the following quadratic form is positive definite: \[ f=5 x_{1}^{2}+x_{2}^{2}+\lambda x_{3}^{2}+4 x_{1} x_{2}-2 x_{1} x_{3}-2 x_{2} x_{3} . \]

1.8.2 Quadratic forms

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Problem: Using the Lagrange method, bring the quadratic form \( x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1} \) to normal form and find the corresponding non-degenerate transformation of variables.

1.8.3 Quadratic forms

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Problem: Convert the quadratic form to canonical form \( 6 x_{1}^{2}+5 x_{2}^{2}+7 x_{3}^{2}-4 x_{1} x_{2}+4 x_{1} x_{3} \) and find the corresponding orthogonal transformation of the variables.

1.8.4 Quadratic forms

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Problem: For the following forms, find a non-degenerate linear transformation that takes the form \( f(x) \) in the form \( g(y) \) (the desired transformation is defined ambiguously). \[ \begin{array}{l} f(x)=2 x_{1}^{2}+9 x_{2}^{2}+3 x_{3}^{2}+8 x_{1} x_{2}-4 x_{1} x_{3}-10 x_{2} x_{3}, \\ g(y)=2 y_{1}^{2}+3 y_{2}^{2}+6 y_{3}^{2}-4 y_{1} y_{2}-4 y_{1} y_{3}+8 y_{2} y_{3} . \end{array} \]

1.8.5 Quadratic forms

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Problem: Find the normal form of the quadratic form using the Lagrange method: \[ f=x_{1}^{2}+x_{2}^{2}+3 x_{3}^{2}+4 x_{1} x_{2}+2 x_{2} x_{3} . \]

1.8.6 Quadratic forms

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