Question-33

Consider the quadratic function in three variables: \[ f(x, y, z) = 3x^2 - 2y^2 +z^2- 2xy + 4yz -8zx \] If \(u = \begin{bmatrix}x & y & z\end{bmatrix}^T\) and \(f(x, y, z) = u^T Au\), which of the following is/are \(A\)?

• \(\begin{bmatrix} 3 & -1 & -4\\ -1 & -2 & 2\\ -4 & 2 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 3 & 4 & -4\\ -6 & -2 & 3\\ -4 & 1 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 3 & 0 & -4\\ -1 & -2 & 2\\ 0 & 1 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 3 & -2 & 1\\ -1 & 1 & 2\\ -4 & 0 & 1 \end{bmatrix}\)

Answer

• \(\begin{bmatrix} 3 & -1 & -4\\ -1 & -2 & 2\\ -4 & 2 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 3 & 4 & -4\\ -6 & -2 & 3\\ -4 & 1 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 3 & 0 & -4\\ -1 & -2 & 2\\ 0 & 1 & 1 \end{bmatrix}\)

• \(\begin{bmatrix} 3 & -2 & 1\\ -1 & 1 & 2\\ -4 & 0 & 1 \end{bmatrix}\)

Solution

The general expression of a quadratic form in \(3\) variables is given below:

\[ \begin{aligned} f(x, y, z) &= \mathbf{x}^T \mathbf{A} \mathbf{x}\\\\ &= a_{xx} x^2 + a_{yy}y^2 + a_{zz} z^2\\\\ &+ (a_{xy} + a_{yx}) xy + (a_{yz} + a_{zy}) yz + (a_{zx} + a_{xz}) zx \end{aligned} \]

Here \(\mathbf{x} = \begin{bmatrix}x\\y\\z\end{bmatrix}\) and \(\mathbf{A} =\begin{bmatrix}a_{xx} & a_{xy} & a_{xz}\\a_{yx} & a_{yy} & a_{yz}\\a_{zx} & a_{zy} & a_{zz}\end{bmatrix}\)

We see that options (a) and (b) are valid choices for \(A\).