(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine the largest value of [tex]x_{1}^2 + x_{2}^2 + x_{3}^2[/tex] when [tex]x_{1}^2 + 2x_{2}^2 + 2x_{3}^2 + 2x_{1}x_{2} + 2x_{1}x_{3} + 2x_{2}x_{3} = 1[/tex]

2. Relevant equations

Not sufficiently relevant to produce the expected answer.

3. The attempt at a solution

Completing the square, we get [tex]x_{1}^2 + 2x_{2}^2 + 2x_{3}^2 + 2x_{1}x_{2} + 2x_{1}x_{3} + 2x_{2}x_{3} = (x_1 + x_2 + x_3)^2 + x_{2}^2 + x_{3}^2 = 1[/tex].

Assigning [tex]x_{1}' = x_{1} + x_{2} + x_{3}, x_{2}' = x_{2}, x_{3}' = x_{3}[/tex] as a coordinate change (and checking that the determinant of the transformation matrix is non-zero to ensure its validity), we equivalently get [tex]x_{1} = x_{1}' - x_{2}' - x_{3}', x_{2} = x_{2}', x_{3} = x_{3}'[/tex].

Consequently, the problem can be formulated as determining the largest and smallest value of

[tex]x_{1}^2 + x_{2}^2 + x_{3}^2 = (x_{1}' - x_{2}' - x_{3}')^2 + x_{2}'^2 + x_{3}'^2 = x_{1}'^2 + 2x_{2}'^2 + 2x_{3}'^2 - 2x_{1}'x_{2}' - 2x_{1}'x_{3}' + 2x_{2}'x_{3}'[/tex] (1)

when [tex]x_{1}'^2 + x_{2}'^2 + x_{3}'^2 = 1[/tex].

Rewriting (1), we get: [tex]x_{1}'^2 + 2x_{2}'^2 + 2x_{3}'^2 - 2x_{1}'x_{2}' - 2x_{1}'x_{3}' + 2x_{2}'x_{3}' =

\begin{pmatrix}x_{1}' & x_{2}' & x_{3}' \end{pmatrix}

\begin{pmatrix}

1 & -1 & -1 \\

-1 & 2 & 1 \\

-1 & 1 & 2 \end{pmatrix}

\begin{pmatrix}x_{1}' \\ x_{2}' \\ x_{3}' \end{pmatrix} = X'^tAX' = X''^tDX'' = \lambda_{1} x''_{1}^2 + \lambda_{2} x''_{2}^2 + \lambda_{3} x''_{3}^2

[/tex]

by using the spectral theorem, the relation [tex]X' = TX''[/tex] between the bases and the orthogonality of transformation matrices between orthonormal bases. (TandDare obviously the transformation matrix between the bases and the diagonalized matrix in the orthonormal eigenvector base, respectively.)

Solving the characteristic equation [tex]\begin{vmatrix} 1-\lambda & -1 & -1\\-1 & 2-\lambda & 1\\-1 & 1 & 2-\lambda \end{vmatrix} = 0 \iff (\lambda - 1)(\lambda - (2-\sqrt{3}))(\lambda-(2+\sqrt{3})) = 0[/tex]

we get that [tex]X''^t D X'' = x''_{1}^2 + (2-\sqrt{3}) x''_{2}^2 + (2+\sqrt{3}) x''_{3}^2[/tex]

Since [tex]|X'|^2 = x_{1}'^2 + x_{2}'^2 + x_{3}'^2 = x_{1}''^2 + x_{2}''^2 + x_{3}''^2 = 1[/tex] (all bases involved are orthonormal), it follows that the largest value sought is equal to the largest eigenvalue, [tex]2+\sqrt{3}[/tex], and the smallest value sought is equal to the smallest eigenvalue, [tex]2-\sqrt{3}[/tex].

Turns out that the key to the problem doesn't agree. It states that the largest value is [tex]1/(2-\sqrt{3})[/tex] and the smallest is [tex]1/(2+\sqrt{3})[/tex]. If I knew where I purportedly managed to mess up, you would not be reading this sentence. Some assistance, or perhaps a suggestion for a different approach, will be appreciated.

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# Homework Help: Quadratic form (maxima and minima problem)

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