1. **Problem Statement:** Given the covariance matrix $$\Sigma = \begin{pmatrix} 2 & 3 \\ 3 & 7 \end{pmatrix}$$, compute the eigenvalues and the proportion of total variance explained by the first principal component (PC1). 2. **Formula and Explanation:** To find eigenvalues $$\lambda$$ of matrix $$\Sigma$$, solve the characteristic equation: $$\det(\Sigma - \lambda I) = 0$$ where $$I$$ is the identity matrix. For a 2x2 matrix $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, the characteristic polynomial is: $$ (a - \lambda)(d - \lambda) - bc = 0 $$ 3. **Apply to the given matrix:** $$ (2 - \lambda)(7 - \lambda) - 3 \times 3 = 0 $$ $$ (2 - \lambda)(7 - \lambda) - 9 = 0 $$ Expanding: $$ 14 - 2\lambda - 7\lambda + \lambda^2 - 9 = 0 $$ $$ \lambda^2 - 9\lambda + 5 = 0 $$ 4. **Solve quadratic equation:** Using quadratic formula: $$ \lambda = \frac{9 \pm \sqrt{(-9)^2 - 4 \times 1 \times 5}}{2} = \frac{9 \pm \sqrt{81 - 20}}{2} = \frac{9 \pm \sqrt{61}}{2} $$ Calculate $$\sqrt{61} \approx 7.81$$: $$ \lambda_1 = \frac{9 + 7.81}{2} = \frac{16.81}{2} = 8.405 $$ $$ \lambda_2 = \frac{9 - 7.81}{2} = \frac{1.19}{2} = 0.595 $$ 5. **Interpretation:** Eigenvalues represent the variance explained by each principal component. 6. **Proportion of variance explained by PC1:** Total variance = sum of eigenvalues = $$8.405 + 0.595 = 9$$ Proportion explained by PC1: $$ \frac{8.405}{9} \approx 0.934 $$ or 93.4%. **Final answers:** - Eigenvalues: $$\lambda_1 = 8.405$$, $$\lambda_2 = 0.595$$ - Proportion of variance explained by PC1: 93.4%