Final answer:
The condition number under the 2-norm for a symmetric matrix A with negative eigenvalues -1 to -n is equal to n, which is the absolute value of the largest eigenvalue in magnitude.
Step-by-step explanation:
The condition number under the 2-norm for a matrix A ∈ ℝ^{n \times n} that is symmetric with eigenvalues -1, -2, ..., -n can be determined using the formula K_2(A) = ||A||_2 \cdot ||A^{-1}||_2. For symmetric matrices, the 2-norm is equal to the absolute value of the eigenvalue with the largest magnitude.
Since the eigenvalues of A are all negative and we are looking for the largest magnitude, the 2-norm of A is |-n| = n. The matrix A-1 will have eigenvalues the reciprocal of A, thus the eigenvalues of A-1 are -1, -1/2, ..., -1/n. Hence, the 2-norm of A-1 is the reciprocal of the smallest magnitude of A's eigenvalues, which is 1. Therefore, the condition number K_2(A) is simply n * 1 = n.