Question

Find the symmetric matrix for the given quadratic form, \(Q = 2x_1^2 + 2x_2^2 + 5x_3^2 – 3x_1 x_2+7x_3 x_1.\)

(a) \(\begin{bmatrix} 1 & \frac{-7}{2} & \frac{3}{2} \\ \frac{-7}{2} & 2 & 0 \\ \frac{3}{2} & 0 & \frac{5}{2}\end{bmatrix} \)

(b) \(\begin{bmatrix} 1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix} \)

(c) \(\begin{bmatrix} 1 & \frac{3}{2} & \frac{-7}{2}\\ \frac{3}{2} & 2 & 0 \\ -\frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix} \)

(d) \(\begin{bmatrix} 1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & 5\end{bmatrix} \)

The question was asked in quiz.

My doubt stems from Real Matrices: Symmetric, Skew-symmetric, Orthogonal Quadratic Form topic in division Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

The correct choice is (b) \(\begin{bmatrix} 1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix} \)

To explain I would say: The following steps need to be followed to obtain the symmetric matrix:

Step 1: There are three variables in the given quadratic equation. Hence, the symmetric matrix to be formed should be of dimension 3×3 and the general form can be written as,

 Q = \(\begin{bmatrix}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} \)

Step 2: Place the square term coefficients of the quadratic equation (2, 2, 5) on the diagonal of the matrix.

Q = \(\begin{bmatrix}2 & c_{12} & c_{13} \\ c_{21} & 2 & c_{23} \\ c_{31} & c_{32} & 5\end{bmatrix} \)

Step 3: Place the remaining coefficients of \(x_i x_j \,at\, c_{ij},\) i.e. coefficient of x1 x2  (-3) at c12 and so on.

Q = \(\begin{bmatrix}2&-3 & 0 \\ 0 & 2 & 0 \\ 7 & 0 & 5\end{bmatrix} \)

Step 4: For a symmetric matrix, S = \(\frac{1}{2} (Q+ Q^T)\)

S = \(\frac{1}{2} \begin{bmatrix}2&-3 & 0 \\ 0 & 2 & 0 \\ 7 & 0 & 5\end{bmatrix} + \begin{bmatrix}2 & 0 & 7 \\ -3 & 2 & 0 \\ 0 & 0 & 5\end{bmatrix} \)

\(S = \frac{1}{2} \begin{bmatrix} 2 & -3 & 7 \\ -3 & 4 & 0 \\ 7 & 0 & 10\end{bmatrix} \)

S = \(\begin{bmatrix}1 & \frac{-3}{2} & \frac{7}{2} \\ \frac{-3}{2} & 2 & 0 \\ \frac{7}{2} & 0 & \frac{5}{2}\end{bmatrix} \)