🧊 Linear Transformation 3D Visualizer

Type any 3×3 matrix and watch it smoothly warp space: the unit cube becomes a parallelepiped whose volume is the determinant, and real eigenvectors light up as the lines that only stretch. Computes the determinant, inverse, rank, trace, eigenvalues, singular values and condition number.

drag to orbit · scroll / pinch to zoom · double-click resets

Matrix A (columns = where x̂, ŷ, ẑ land)

Morph I → A 100%

Grid lattice Eigenvectors Original cube Unit sphere → ellipsoid

Test vector v (x, y, z)

Determinant (volume scale)—

Trace—

Rank—

Eigenvalues—

Orientation—

Type—

Singular values σ—

Condition number σ₁/σ₃—

Char. polynomial—

Eigenvectors—

Rotation part—

A·v—

Inverse A⁻¹—

How this works — reading a matrix geometrically

A 3×3 matrix is a machine that moves every point of space: v ↦ A v. Its three columns are exactly where the basis vectors x̂ (red), ŷ (green) and ẑ (blue) land, and every other point follows because the map is linear — grid lines stay straight, parallel and evenly spaced, and the origin never moves.

Determinant — the signed volume of the transformed unit cube. |det| > 1 expands space, < 1 shrinks it, 0 flattens it onto a plane or line (no inverse), and a negative value means space was mirrored (orientation flipped).
Eigenvectors — directions that stay on their own line: A v = λ v. The eigenvalues are the roots of the characteristic cubic λ³ − tr(A) λ² + c₂ λ − det(A) = 0, solved here in closed form (complex pairs mean a rotation part).
Inverse — computed from the adjugate: A⁻¹ = adj(A) / det(A).
Morph slider — shows (1−t) I + t A, a straight-line blend from “do nothing” to A. It is a visual aid, not a path through rotations.

Worked example: the shear [[1,1,0],[0,1,0],[0,0,1]] tilts the cube but keeps its volume (det = 1); its only eigen-direction is x̂, with λ = 1 repeated three times.