Matrix Transform Visualizer & Eigenvalue Explorer

Drag basis vector heads î (Cyan) or ĵ (Coral) directly

Scale: 1 unit = 50px

Presets:

Transformation Matrix A

det(A) = 3.00

A =

a (î_x)

b (ĵ_x)

c (î_y)

d (ĵ_y)

a: 2.0

b: 1.0

c: 1.0

d: 2.0

Eigenspace Analysis

SVD Breakdown (A = U Σ Vᵀ)

Decomposes any transformation into pure rotation Vᵀ, diagonal scaling Σ, and final rotation U.

Geometric Interpretation

The matrix acts on space by scaling along symmetric eigen-axes. Areas are scaled by a factor of 3.00, preserving orientation.

Matrices move basis vectors, area and eigenlines

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Read the explanation

A matrix tells you where the two basis vectors go. This tool starts with two one, one two. The cyan first column moves the unit horizontal vector to two one. The coral second column moves the vertical unit vector to one two. Every input vector is a weighted sum of those two columns. Here one unit is eighty pixels, so the arrow coordinates encode the actual numbers, rather than decorative motion. The unit square becomes a parallelogram. Its corner coordinates are zero zero, two one, three three, and one two. The signed determinant is two times two minus one times one, which is three. The new polygon therefore has three times the original area. The singular projection preset instead makes the two columns dependent, collapsing the area to zero. The geometry is the calculation you can test with the determinant badge. Two special directions do not rotate under this matrix. Along one one, the output is three three: an eigenvalue of three. Along one negative one, the output stays one negative one: an eigenvalue of one. The amber diagonal arrow triples in length, while the other direction remains unchanged. Use the rotation preset to see why real eigenlines can disappear, then export the actual matrix and metrics from the tool. This explanation does not prove every export or drag interaction works until the paired native checks pass.