Eigenvector Spotter - Linear Algebra Practice

Given a 2x2 matrix and four candidate directions, click the one that A merely stretches instead of rotating - the eigenvector.

Each round the game builds a random 2x2 matrix A = P·D·P⁻¹ from a random invertible integer matrix P and a diagonal D with two distinct nonzero integer eigenvalues. By construction the columns of P are exactly the eigenvectors of A. You are shown A (entries displayed rounded to 2 decimals, computed at full precision) and four labeled arrows on a small plot: one true eigenvector and three decoy directions, each checked at generation time to not be an eigenvector and to not be parallel to either true eigenvector.

Click the direction that stays on its own line when A is applied - the one A only stretches or flips. After you pick, the reveal shows A·v for your choice and states whether the result is a scalar multiple of the input. The correctness test in the code is the cross product: v is an eigenvector when (Av)ₓ·vᵧ - (Av)ᵧ·vₓ is zero (within 1e-6), meaning Av and v are parallel.

There is one correct option per round - the puzzle deliberately excludes the other eigenvector from the candidates. Press next for a fresh matrix; rounds continue indefinitely.

Why quant interviews test this

Quant-research and quant-dev linear algebra screens almost always touch eigenvectors: define them, find them for a small matrix, explain diagonalization, or connect them to PCA on a covariance matrix. The 2x2 case is the standard whiteboard size, and interviewers watch whether you reach for the definition Av = λv and a parallelism check before grinding the characteristic polynomial.

Follow-ups this game prepares you for: what happens when eigenvalues repeat (the identity-multiple degenerate case), why symmetric matrices have orthogonal eigenvectors, and why the top eigenvector of a covariance matrix is the direction of maximal variance.

The Eigenvector Spotter guide covers how scoring works, the strategy that wins, a worked example and the mistakes most players make.

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