Find minimal a for PSD of 3x3 matrix
Company: Capital One
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Technical Screen
Find the smallest real a such that the symmetric matrix with ones on the diagonal and a on all off-diagonal entries, A = [[1, a, a], [a, 1, a], [a, a, 1]], is positive semidefinite. Justify using eigenvalues (or Sylvester’s criterion), showing all steps.
Quick Answer: This question evaluates understanding of linear algebra concepts such as eigenvalues, matrix definiteness, and spectral properties by asking for the minimal real parameter a that makes a symmetric 3×3 matrix with ones on the diagonal and constant off-diagonal entries positive semidefinite.
