Socratic Matrix & Eigenvector Visual Explorer

Vector v: (1.00, 1.00)

Transformed T(v): (3.00, 1.00)

Eigen-lines: 2 Real

Transformation Presets

a (î_x): 2.0

b (ĵ_x): 1.0

c (î_y): 0.0

d (ĵ_y): 1.0

Socratic Math Dialogue

Stage 1/3

Your Intuition Hypothesis:

Computed Spectral Properties

Eigenvalues (λ): λ1 = 2.00, λ2 = 1.00

Eigenvectors (v): [1, 0], [1, -1]

Span Invariance: Span Preserved on 2 Lines

Canonical Proof & Discovery Summary

Horizontal Shear preserves vector direction along the x-axis (y = 0)

Under matrix T = [[2, 1], [0, 1]], vectors on the span of [1, 0] scale by eigenvalue λ = 2 without changing direction.

Socratic Mastery Level

Intuition Unlocked

Find the invariant line instead of guessing

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Read the explanation

The source shear preset uses matrix rows two one and zero one. Its selected vector one one becomes three one. That vector changes direction. But vector one zero becomes two zero, staying on the horizontal line with eigenvalue two. The other eigenvalue is one, with direction one minus one. The animated arrows distinguish a general vector from an invariant line. The source rotation preset is the exact quarter turn matrix zero minus one, one zero. It maps one zero to zero one, so a real nonzero vector cannot stay on its original line. Trace is zero and determinant one, giving characteristic equation lambda squared plus one equals zero. Its roots are complex plus and minus i. The source distinguishes this from presets with real eigenvectors. The singular preset is rows one one and one one. Its trace two and determinant zero give eigenvalues two and zero. Vector one one stretches by two, while one minus one maps to zero. The guided dialogue is a saved teaching curriculum, not a live model inference. Matrix controls and the original exported image illustrate these source computations, without proving properties of arbitrary higher dimensional systems.