Find minimal a for PSD of 3x3 matrix

Q: Find minimal a for PSD of 3x3 matrix

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.

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Question

PSD threshold for a 3×3 matrix with constant off-diagonal entries

You are given the symmetric matrix with ones on the diagonal and a constant real value a on all off-diagonal entries:

A = [[1, a, a], [a, 1, a], [a, a, 1]].

Task: Find the smallest real number a such that A is positive semidefinite (PSD). Justify your answer using eigenvalues (or Sylvester’s criterion), and show all steps.

Find minimal a for PSD of 3x3 matrix

PSD threshold for a 3×3 matrix with constant off-diagonal entries

Solution

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Find minimal a for PSD of 3x3 matrix

Overview

PSD threshold for a 3×3 matrix with constant off-diagonal entries

Solution

Comments (0)