Eigenvalue and Eigenvector Calculator

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvalue (λ) is a scalar that, when multiplied by a vector, produces the same result as multiplying the vector by a matrix. An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. Mathematically, for a matrix A and eigenvector v: Av = λv.

To find eigenvalues and eigenvectors: 1) Calculate the characteristic polynomial det(A - λI) = 0, 2) Solve for the roots of this polynomial to get eigenvalues, 3) For each eigenvalue, solve the system (A - λI)v = 0 to find the corresponding eigenvectors. Our calculator automates this process for matrices up to 6x6.

The characteristic polynomial is det(A - λI), where A is the matrix, λ is the eigenvalue variable, and I is the identity matrix. The roots of this polynomial are the eigenvalues of the matrix. For example, for a 2x2 matrix, the characteristic polynomial is λ² - (trace)λ + (determinant) = 0.

Eigenvalues and eigenvectors are crucial in many applications: diagonalization of matrices, solving systems of differential equations, principal component analysis (PCA) in data science, quantum mechanics, vibration analysis in engineering, and many other fields in physics, engineering, and computer science.

Real eigenvalues are scalar values that are real numbers, while complex eigenvalues have both real and imaginary parts. Real eigenvalues correspond to real eigenvectors, while complex eigenvalues come in conjugate pairs and correspond to complex eigenvectors. Our calculator focuses on real eigenvalues and eigenvectors.

The calculator shows: 1) The eigenvalues (λ values) with their multiplicities, 2) The corresponding eigenvectors as column vectors, 3) The characteristic polynomial, and 4) Step-by-step solution process. Each eigenvalue-eigenvector pair satisfies the equation Av = λv.

The multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial. Algebraic multiplicity counts repeated roots, while geometric multiplicity is the dimension of the eigenspace (number of linearly independent eigenvectors) for that eigenvalue.

Yes, some matrices have only complex eigenvalues. For example, rotation matrices in 2D have complex eigenvalues. Our calculator is designed for matrices with real eigenvalues. If your matrix has complex eigenvalues, you may need specialized software for complex arithmetic.

Eigenvalues reveal important matrix properties: the sum of eigenvalues equals the trace, the product equals the determinant, eigenvalues of A⁻¹ are reciprocals of A's eigenvalues, and eigenvalues of Aⁿ are the nth powers of A's eigenvalues. Zero eigenvalues indicate singular matrices.

A matrix is diagonalizable if it has n linearly independent eigenvectors (where n is the matrix size). The diagonal matrix contains eigenvalues on the diagonal, and the transformation matrix contains eigenvectors as columns. This simplifies matrix operations and powers.

Eigenvalues are used in: structural analysis (natural frequencies), image processing (singular value decomposition), machine learning (principal component analysis), quantum mechanics (energy levels), population dynamics, and many other scientific and engineering applications.

Our calculator uses numerical methods with high precision (typically 10⁻¹⁰ tolerance). For exact symbolic calculations, specialized computer algebra systems may be preferred. The accuracy is sufficient for most practical applications in science and engineering.