Matrix Transformer

Each challenge checks itself. Work on this keeps it in view above the controls.

Set a 2x2 or 3x3 matrix and watch it act on the unit square, circle, cube or sphere: the columns, the signed area or volume (determinant), the ellipse half-axes (singular values), and the motion from I to A.

How to use

• Type the four entries of A or load an example (rotation, reflection, shear, projection, scaling).
• The first column of A is where e₁ = (1, 0) lands (solid arrow A e₁); the second column is where e₂ = (0, 1) lands (dashed arrow A e₂). Everything else follows by linearity.
• The unit square (shaded gold, dashed outline: before) becomes the blue parallelogram (after). Its area is |det A|. If det A < 0 the parallelogram is also hatched and marked "flipped": the transformation reverses orientation, like a mirror.
• Choose the unit circle to see that A turns every circle into an ellipse: its half-axes are the singular values of A (the square roots of the eigenvalues of A^TA) and its area is π|det A|.
• Space (3 × 3). Type a 3 × 3 matrix or load an example. The unit cube (gold, dashed) becomes a parallelepiped (blue) spanned by the columns Ae₁, Ae₂, Ae₃; its volume is |det A|. The unit sphere becomes an ellipsoid. Turn and tilt the view, or drag it.
• Use the t slider or the Animate button to watch the plane move from where it starts (t = 0) to its image (t = 1). Animation is skipped if your system asks for reduced motion.
• Keyboard: Tab between controls, arrow keys to change numbers and the slider. The status box is read aloud by screen readers.

Accessibility: every control is a standard form control you can reach with Tab and change with the arrow keys; the status box is announced by screen readers; the picture has a text description (press Describe) that updates as you work; animations are skipped when your system asks for reduced motion.