Eigenvalue and Eigenvector Calculator
Calculate eigenvalue eigenvector instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Eigenvalue and Eigenvector Calculator
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Formula: det(A - lambda * I) = 0 (characteristic equation), then solve (A - lambda * I)v = 0 for eigenvectors
Worked example — Eigenvalues: 1, 2, 4 | Matrix is symmetric, so orthogonally diagonalizable
Formula
det(A - lambda * I) = 0 (characteristic equation), then solve (A - lambda * I)v = 0 for eigenvectors
Eigenvalues are found by solving the characteristic equation, which sets the determinant of (A - lambda * I) to zero. For a 3x3 matrix, this produces a cubic polynomial. Each eigenvalue lambda has corresponding eigenvectors found by solving the homogeneous system (A - lambda * I)v = 0.
Worked Examples
Example 1: 3x3 Symmetric Matrix Eigenvalues
Problem:Find eigenvalues and eigenvectors of A = [[2, 1, 0], [1, 3, 1], [0, 1, 2]].
Solution:Characteristic polynomial: lambda^3 - 7*lambda^2 + 14*lambda - 8 = 0 Factors: (lambda - 1)(lambda - 2)(lambda - 4) = 0 Eigenvalues: lambda = 1, 2, 4 For lambda = 1: [1, 1, 0; 1, 2, 1; 0, 1, 1]v = 0 -> v1 = [1, -1, 1] For lambda = 2: [0, 1, 0; 1, 1, 1; 0, 1, 0]v = 0 -> v2 = [1, 0, -1] For lambda = 4: [-2, 1, 0; 1, -1, 1; 0, 1, -2]v = 0 -> v3 = [1, 2, 1]
Result:Eigenvalues: 1, 2, 4 | Matrix is symmetric, so orthogonally diagonalizable
Example 2: Matrix with Repeated Eigenvalue
Problem:Find eigenvalues of A = [[3, 0, 0], [0, 3, 0], [0, 0, 5]].
Solution:This is a diagonal matrix, so eigenvalues are the diagonal entries. lambda1 = lambda2 = 3 (algebraic multiplicity 2) lambda3 = 5 Eigenvectors for lambda = 3: any vector in the xy-plane, v1 = [1,0,0], v2 = [0,1,0] Eigenvector for lambda = 5: v3 = [0,0,1] Geometric multiplicity = algebraic multiplicity, so diagonalizable.
Result:Eigenvalues: 3, 3, 5 | Diagonalizable (geometric multiplicity = algebraic multiplicity)
Frequently Asked Questions
What are eigenvalues and eigenvectors?
Eigenvalues and eigenvectors are fundamental concepts in linear algebra that describe the behavior of a linear transformation. An eigenvector v of a matrix A is a nonzero vector that, when multiplied by A, only gets scaled by a factor called the eigenvalue lambda: Av = lambda * v. The eigenvector direction is preserved by the transformation, only its magnitude changes. Eigenvalues can be positive (stretching in the same direction), negative (reversing direction), zero (collapsing to zero), or complex (indicating rotation). Every square matrix has eigenvalues, though they may be complex even for real matrices.
How do you find eigenvalues of a matrix?
Eigenvalues are found by solving the characteristic equation det(A - lambda*I) = 0, where I is the identity matrix. For a 3x3 matrix, this produces a cubic polynomial in lambda. The coefficients of this polynomial are related to the trace (sum of diagonal entries), the sum of 2x2 cofactors along the diagonal, and the determinant of A. The roots of this cubic equation are the eigenvalues. For 2x2 matrices, the quadratic formula suffices. For larger matrices, numerical methods like the QR algorithm are used because there is no general formula for polynomials of degree 5 or higher by the Abel-Ruffini theorem.
How do you find eigenvectors once you know the eigenvalues?
For each eigenvalue lambda, the corresponding eigenvectors are found by solving the homogeneous system (A - lambda*I)v = 0. This means finding the null space of the matrix (A - lambda*I). Row reduce this matrix to echelon form and identify the free variables. Each free variable gives rise to one linearly independent eigenvector. The collection of all eigenvectors for a given eigenvalue, together with the zero vector, forms the eigenspace. The dimension of the eigenspace is called the geometric multiplicity of the eigenvalue. Any nonzero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.
What is the relationship between eigenvalues and the trace and determinant?
There are elegant relationships between eigenvalues and matrix properties. The sum of all eigenvalues equals the trace (sum of diagonal elements) of the matrix. The product of all eigenvalues equals the determinant of the matrix. For a 3x3 matrix with eigenvalues lambda1, lambda2, lambda3: trace = lambda1 + lambda2 + lambda3 and det = lambda1 * lambda2 * lambda3. These relationships provide quick checks for computed eigenvalues and also give insight into the matrix without computing eigenvalues explicitly. A zero determinant means at least one eigenvalue is zero, indicating the matrix is singular.
What are some real-world applications of eigenvalues and eigenvectors?
Eigenvalues and eigenvectors appear throughout science and engineering. In structural engineering, they determine natural frequencies and vibration modes of buildings and bridges. Google PageRank algorithm uses the dominant eigenvector of the web link matrix to rank web pages. In quantum mechanics, eigenvalues represent measurable quantities like energy levels. Principal Component Analysis (PCA) in machine learning uses eigenvectors of the covariance matrix for dimensionality reduction. In image compression, eigenvalues of the image matrix identify the most important features. Stability analysis of dynamical systems depends entirely on the eigenvalues of the system matrix.
What does it mean if eigenvalues are complex?
Complex eigenvalues indicate that the linear transformation involves rotation. A real matrix can have complex eigenvalues, which always come in conjugate pairs (a + bi and a - bi). The real part determines growth or decay, and the imaginary part determines the frequency of rotation. For example, a pure rotation matrix has eigenvalues on the unit circle (|lambda| = 1) with zero real part growth. In dynamical systems, complex eigenvalues produce oscillatory behavior. If the magnitude is less than 1, the oscillation decays; if greater than 1, it grows. Systems with complex eigenvalues near the imaginary axis are at the boundary between stability and instability.
What is the spectral theorem and why is it important?
The spectral theorem states that every real symmetric matrix can be diagonalized by an orthogonal matrix, with all eigenvalues being real. This means A = QDQ^T, where Q is orthogonal (Q^T = Q^(-1)) and D is diagonal. The columns of Q are orthonormal eigenvectors. This theorem is fundamental because it guarantees nice properties for symmetric matrices that arise naturally in many applications. Covariance matrices, Hessian matrices in optimization, adjacency matrices of undirected graphs, and stress tensors in physics are all symmetric. The spectral theorem ensures their eigendecomposition always exists and has real eigenvalues.
How do eigenvalues determine the stability of a system?
In dynamical systems described by dx/dt = Ax, the eigenvalues of A completely determine the stability of the equilibrium point at the origin. If all eigenvalues have negative real parts, the system is asymptotically stable (all trajectories converge to zero). If any eigenvalue has a positive real part, the system is unstable (some trajectories diverge). If eigenvalues are purely imaginary, the system exhibits neutral stability with periodic oscillations. For discrete-time systems x(n+1) = Ax(n), stability requires all eigenvalues to have magnitude less than 1. This eigenvalue-based stability analysis is the cornerstone of control theory and is used to design stable feedback systems.
What is the difference between algebraic and geometric multiplicity?
The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace (the number of linearly independent eigenvectors for that eigenvalue). The geometric multiplicity is always less than or equal to the algebraic multiplicity. When they are equal for every eigenvalue, the matrix is diagonalizable. When the geometric multiplicity is strictly less than the algebraic multiplicity for some eigenvalue, the matrix is defective and cannot be diagonalized. For example, the matrix [[2, 1], [0, 2]] has eigenvalue 2 with algebraic multiplicity 2 but geometric multiplicity 1.
Can a matrix have repeated eigenvalues?
Yes, a matrix can have repeated (degenerate) eigenvalues. The identity matrix has eigenvalue 1 with multiplicity n for an n x n matrix. Repeated eigenvalues are common in matrices with symmetry. Whether a matrix with repeated eigenvalues is diagonalizable depends on the geometric multiplicity. If the geometric multiplicity equals the algebraic multiplicity, the matrix is still diagonalizable (like the identity matrix, which is already diagonal). If geometric multiplicity is less, the matrix is defective and requires Jordan normal form instead of diagonalization. Symmetric matrices with repeated eigenvalues are always diagonalizable because the spectral theorem guarantees sufficient eigenvectors.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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