Eigenvector Safety Matrix Lab

💡 Drag vector tip directly on grid | Gold lines = Invariant Eigen-Directions

Transformation Matrix A det(A) = 2.0

a (x scale) 2.0

b (x shear) 1.0

c (y shear) 0.0

d (y scale) 1.0

EIGENVALUE / EIGENVECTOR SPACE

Vector Transformation Breakdown

SAFE INVARIANT

v (initial): [1.00, 2.00]

A·v (transformed): [4.00, 2.00]

Angular deflection: 26.6°

Live Diagnostic Proof

Vector Alignment: OFF-AXIS

Trapped in Shear: YES

Scale Factor (λ): N/A

Directions that survive a matrix transformation

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Read the explanation

The saved matrix begins with first row two, one and second row zero, one. Applied to vector one, two, multiplication gives four, two. The arrow endpoints show those actual coordinate pairs on one scale: the original points toward one, two while its image points toward four, two. A change in direction means this vector is not an eigenvector of this matrix, even though both arrows have positive components. For the same matrix, vector one, zero transforms to two, zero. The direction is unchanged and the scale factor is two, so it is an eigenvector with eigenvalue two. Vector one, minus one transforms to one, minus one and has eigenvalue one. The source computes eigenvalues from trace and determinant: trace three, determinant two, so the quadratic roots are two and one. The rotation preset uses first row zero, minus one and second row one, zero. Its vector two, one becomes minus one, two, a ninety-degree turn. Trace is zero and determinant one, so the eigenvalue discriminant is minus four. The real solver therefore returns no real eigenvectors for this preset. The interface calls invariant directions safe, but that is a geometric label in this toy matrix lab, not a claim that any real system is safe.