Eigenvalue and Eigenvector Calculator

An eigenvalue and eigenvector describe a direction that a linear transformation does not rotate away from itself. When a square matrix multiplies an eigenvector, the result points along the same line as the original vector. The eigenvalue is the scale factor applied in that direction. A positive eigenvalue preserves direction, a negative eigenvalue reverses it, and a zero eigenvalue collapses that direction to the zero vector. These ideas make complicated matrix transformations easier to understand through a few important directions and scale factors. For a 2 by 2 matrix, the calculator forms the characteristic polynomial from the trace and determinant. Solving that quadratic equation produces up to two real eigenvalues. For each eigenvalue, it solves the homogeneous system created by subtracting that eigenvalue from the diagonal. The displayed eigenvector is normalized so its Euclidean length is one. An eigenvector can be multiplied by any nonzero constant and remain valid, so your textbook or software may show an equivalent vector with different signs or magnitudes. Eigenvalues appear throughout linear algebra, differential equations, statistics, data science, physics, engineering, and computer graphics. They reveal stable and unstable behavior in dynamic systems, principal directions in covariance matrices, vibration modes in structures, and long-term patterns in repeated matrix multiplication. Distinct eigenvalues of a 2 by 2 matrix generally provide two independent eigenvectors and make the matrix diagonalizable. A repeated eigenvalue may provide either one or two independent directions, depending on the matrix. This calculator focuses on real eigenvalues for 2 by 2 matrices and reports when the characteristic polynomial has a negative discriminant. Complex eigenvalues are mathematically valid, but they require complex-number notation beyond this focused tool. Decimal results are rounded for display, while calculations use full JavaScript floating-point precision. For exact classroom work, use the displayed result as a numerical check and retain radicals or fractions when the characteristic equation factors neatly. Always verify an eigenpair by multiplying the original matrix by the vector and comparing that product with the eigenvalue times the vector.