Correlation Matrix PSD Visualizer
Find all real values of ρ for which Σ is a valid correlation matrix, i.e. positive semidefinite.
1. Matrix
10.6−0.30.61ρ−0.3ρ1
−1.201.2
2. PSD checklist
Diagonal entries1 ≥ 0
Minor {1,2}1−0.6² = 0.64
Minor {1,3}1−(−0.3)² = 0.91
Minor {2,3}
det(Σ)
det(Σ) = 0.55 − 0.36ρ − ρ²
3. Binding condition: det(Σ) ≥ 0
The determinant is a downward-opening parabola. PSD holds between its two roots.
Solve determinant inequality
ρ² + 0.36ρ − 0.55 ≤ 0
Roots
ρ = (−9 ± 4√91)/50
Final interval
−0.943151… ≤ ρ ≤ 0.583151…
Interview shortcut: for a symmetric 3×3 matrix, PSD is equivalent to all principal minors being nonnegative. The fixed 2×2 minors are already positive, while 1−ρ² ≥ 0 gives |ρ|≤1. The determinant is tighter, giving ρ ∈ [(−9−4√91)/50, (−9+4√91)/50].