ΣMatrix SpaceOpen Notes on Linear Algebra

Interactive Demos

Two self-contained experiments · everything runs locally in your browser, nothing is uploaded

1 Linear transformation explorer

Drag the four sliders to set the entries of

A = [ a b ; c d ]

The faint grid is the original plane; the blue grid is its image under A. The orange arrow is where î lands, the purple arrow where ĵ lands, and the shaded parallelogram is the image of the unit square — its signed area is the determinant. The readout computes the eigenvalues from the trace and determinant: a negative discriminant means the map rotates as well as scales.

det = 0.85  λ₁ = 1.281, λ₂ = 0.719  (real eigenvalues)
Things to try
Set b = c = 0: the grids stay rectangular and the eigenvectors point along the axes. Set a = 0, b = 1, c = −1, d = 0: a pure 90° rotation — the determinant is 1, the eigenvalues are complex. Now approach ad − bc = 0: the plane folds flat onto a line and the matrix becomes singular.

2 Power iteration

The demo runs 24 steps of the power method with Rayleigh-quotient eigenvalue estimates on the fixed symmetric matrix

A = [ 7 2 ; 2 4 ], λ₁ = (11 + √37)/2 ≈ 8.5414.

Because A is symmetric, the Rayleigh quotient converges quadratically once the iterate is close — watch the error curve plunge several decades in the last iterations. Reseed for a different random start vector; the shape of the curve barely changes, which is the point of the convergence theory.

Further reading

The theory behind both demos is developed in the notes on eigenvalues and spectral theory; the geometric dictionary between matrices and plane deformations is animated beautifully in the video library.