Fix $j\in A$ and write
$$
R=\lambda I+X_{A\setminus\{j\}}X_{A\setminus\{j\}}^T,
\qquad a=X_j^TR^{-1}X_j\geq0.
$$
The matrix $R$ is <positive-definite matrix>[positive definite]. Applying the <Sherman–Morrison formula> to $R+X_jX_j^T$ yields
$$
X_j^T(X_AX_A^T+\lambda I)^{-1}X_j
=a-\frac{a^2}{1+a}=\frac a{1+a}<1.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 5 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-205/5/b/solution