Eigenvector Calculator

Because any nonzero scalar multiple of an eigenvector is also an eigenvector. If [1, 1] is an eigenvector, so is [2, 2], [100, 100], or [-3, -3]. The calculator returns one representative direction.

The system (A - lambda I)v = 0 will only have the trivial solution v = 0 if lambda is not actually an eigenvalue. Use the eigenvalue calculator first to find valid eigenvalues, then use this tool.

The same formula applies: solve (A - lambda I)v = 0 using complex arithmetic. The eigenvector components will be complex numbers. Complex eigenvectors always come in conjugate pairs.

The eigenspace for eigenvalue lambda is the set of all vectors v such that Av = lambda v, including the zero vector. It is a subspace of the vector space. For a 2x2 matrix, the eigenspace is either a line through the origin (1-dimensional) or the entire plane (2-dimensional, only when A = lambda I).

In structural engineering, eigenvectors represent vibration modes. In computer vision, they define principal components of image data (eigenfaces). In quantum mechanics, they represent measurable states. In Google PageRank, the dominant eigenvector ranks web pages.

No. If Av = lambda1 v and Av = lambda2 v for the same nonzero v, then lambda1 must equal lambda2. Distinct eigenvalues always have linearly independent eigenvectors.

Yes. Click the "Embed" button at the top of this page to customize the size, colors, and theme, then copy the iframe code. Paste it into any HTML page, WordPress site, or CMS. It is completely free, requires no signup, and works on all devices. You can also visit our embed guide at calculory.com/services/embed-calculators for more details.