Eigenvalue and Eigenvector Calculator

An eigenvalue (̴λ) and its corresponding eigenvector (̴v) satisfy the fundamental linear algebra equation: A̴v = ̴Λ̴v, where A is a square matrix. To find eigenvalues, we compute the characteristic polynomial by taking the determinant of the matrix minus ̴λ times the identity matrix: det(A - ̴Λ I) = 0. For a 2x2 matrix, this expands into a quadratic equation based on the matrix trace and determinant. For a 3x3 matrix, it yields a cubic equation. Once eigenvalues are identified, substituting each back into (A - ̴Λ I)̴v = 0 allows us to solve the homogeneous linear system for the respective eigenvectors.

Let us find the eigenvalues and eigenvectors for a simple 2x2 matrix A, where the top row is [2, 1] and the bottom row is [1, 2]. First, we compute the trace (sum of diagonal elements: 2 + 2 = 4) and the determinant (product of diagonals minus anti-diagonals: (2 * 2) - (1 * 1) = 3). Setting up the characteristic equation det(A - ̴Λ I) = 0 gives us ̴Λ^2 - 4̴Λ + 3 = 0. Factoring this quadratic expression yields (̴Λ - 3)(̴Λ - 1) = 0, meaning our eigenvalues are ̴Λ_1 = 3 and ̴Λ_2 = 1. Next, for ̴Λ_1 = 3, we plug it into (A - 3I)̴v = 0, leading to the system -x + y = 0, which gives the eigenvector [1, 1]. Similarly, for ̴Λ_2 = 1, we get the eigenvector [1, -1].

Always verify that your input matrix is square (such as 2x2 or 3x3), as eigenvalues and eigenvectors are only defined for square matrices. When dealing with fractional or decimal entries, double-check your arithmetic during the characteristic polynomial expansion to avoid downstream errors in your eigenvector solutions.