The transformation ellipse shown in the accompanying diagram represents the quadratic form $\mathbf{x}^T A \mathbf{x} = 1$, where matrix $A$ is a real $2 \times 2$ symmetric positive-definite matrix. The principal axes are aligned along the orthogonal eigenvectors $\mathbf{v}_1$ and $\mathbf{v}_2$ with eigenvalues $\lambda_1 = 4$ and $\lambda_2 = 1$. Which of the following statements are necessarily TRUE?
Options:
(A) The semi-major axis of the ellipse has length $1/\sqrt{\lambda_2} = 1.0$
(B) The semi-minor axis of the ellipse has length $1/\sqrt{\lambda_1} = 0.5$
(C) The eigenvectors $\mathbf{v}_1$ and $\mathbf{v}_2$ are mutually perpendicular
(D) The determinant of matrix $A$ is equal to 4
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. For the quadratic form $\mathbf{x}^T A \mathbf{x} = 1$, rotating to the principal axis basis yields $\lambda_1 y_1^2 + \lambda_2 y_2^2 = 1 \implies \frac{y_1^2}{(1/\sqrt{\lambda_1})^2} + \frac{y_2^2}{(1/\sqrt{\lambda_2})^2} = 1$.
2. The semi-axis lengths are $a = 1/\sqrt{\lambda_2} = 1/\sqrt{1} = 1.0$ (semi-major) and $b = 1/\sqrt{\lambda_1} = 1/\sqrt{4} = 0.5$ (semi-minor). (Statements A and B True)
3. For any real symmetric matrix, eigenvectors corresponding to distinct eigenvalues are mutually orthogonal ($\mathbf{v}_1 \perp \mathbf{v}_2$). (Statement C True)
4. $\det(A) = \lambda_1 \lambda_2 = 4 \times 1 = 4$. (Statement D True)