Eigenvalue Eigenvector Calculator

What Are Eigenvalues and Eigenvectors?

For a square matrix $A$, an eigenvector $\mathbf{v}$ and eigenvalue $\lambda$ satisfy $A\mathbf{v} = \lambda\mathbf{v}$. Eigenvalues reveal scaling behavior; eigenvectors show the directions that stay aligned under the transformation.

Characteristic Equation

For a $2 \times 2$ matrix this yields a quadratic polynomial. For $3 \times 3$, a cubic polynomial whose roots are the eigenvalues.

Finding Eigenvectors

For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{v} = \mathbf{0}$ using row reduction. The resulting nonzero vector (normalized) is an eigenvector.

Related tools: Matrix Determinant Calculator and Column Space Calculator.