Norms and Inner Products: Unit Balls

Is A positive definite?

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

See the unit balls of the 1-, 2- and max-norms and of the A-norm side by side, with the length of your own vector x in each; find out when a symmetric matrix A gives a norm (positive definiteness).

How to use

• A unit ball of a norm is the set of vectors of length at most 1 in that norm; its edge is the unit circle of the norm. The 1-norm gives a diamond, the 2-norm a round disc, the max-norm a square. For a symmetric positive definite (SPD) matrix A, the A-norm ‖x‖_A = √(x^TAx) gives an ellipse.
• Type your vector x, or click or drag in any picture. Each picture shows x, its unit ball, and the point x/‖x‖ where the ray through x leaves the ball. The caption above each picture gives the length of x in that norm.
• The same vector has different lengths in different norms: ‖x‖_∞ ≤ ‖x‖₂ ≤ ‖x‖₁. The bigger the unit ball in a direction, the shorter x counts in that direction.
• Type the entries a, b, c of the symmetric matrix A. "Is A positive definite?" checks x^TAx > 0 for every x ≠ 0, using a > 0 and det A > 0, the eigenvalues, and completing the square. When A is not positive definite, x^TAx = 1 is not a closed curve and the A-norm is not a norm.
• All four pictures use the same window, so you can compare the balls; "automatic" fits x and the balls.

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.