Algèbre Linéaire Avancée · Paris-Saclay

Normal vs. Non-Normal Matrices

Yehor Korotenko

A normal matrix scales each axis of an orthogonal frame independently. A non-normal matrix shears — directions inevitably bleed into each other.

01 — How the unit circle transforms
Normal Matrix (Hermitian)
A = R(θ)·diag(λ₁,λ₂)·R(θ)ᵀ — eigenvectors always ⊥
2.00
0.50
0.50
Unit circle
Image ellipse
Eigenvectors (⊥)
Non-Normal Matrix
A = [[a,b],[0,d]] — shear causes non-orthogonal eigenvectors
1.50
1.20
0.50
Unit circle
Image ellipse
Eigenvectors (skewed)
Key Observation

For the normal matrix, the ellipse axes always align with the blue eigenvectors — which are always perpendicular. It purely scales each orthogonal direction by λ₁ and λ₂.

For the non-normal matrix, when b ≠ 0 the eigenvectors are skewed. No change of coordinates makes this matrix purely diagonal — the shear is intrinsic and irreducible.

02 — Eigenvector orthogonality
Normal — eigenvectors always 90°
Non-normal — angle shrinks with shear b
03 — Why AA* = A*A captures normality
Normal: order of A and A* doesn't matter

Same eigenbasis → both just scale along fixed axes → scalings commute

x→A→A* = x→A*→A
AA* = A*A ✓

‖Aⁿ‖ = ρ(A)ⁿ — spectrum controls everything

Non-normal: order matters

Shear makes A and A* act on different frames → different results

x→A→A* ≠ x→A*→A
AA* ≠ A*A ✗

‖A‖ can far exceed ρ(A) — eigenvalues mislead

04 — Spectral radius vs. operator norm (shear demo)

Matrix A = [[1.5, b],[0, 0.5]]. Eigenvalues fixed at 1.5, 0.5. Watch ‖A‖ grow while ρ(A) stays constant.

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ρ(A) = 1.5 (constant)
‖A‖ = largest singular value (grows)
One-sentence summary

A matrix is normal when its geometric action decomposes into independent scalings along orthogonal axes — no direction bleeds into another, A and A* don't interfere with each other, and the spectrum completely controls behavior.