Eigenvalue and Eigenvector Calculator
Enter a 2x2 or 3x3 square matrix and this calculator instantly finds every real eigenvalue, the corresponding unit eigenvectors, the characteristic polynomial, trace, determinant, and tells you whether the matrix is diagonalizable. Choose the matrix size, fill in the entries, and the "Show your work" panel walks through each step of the algebra.
What are eigenvalues and eigenvectors?
Given a square matrix A, an eigenvector is a non-zero vector v that only stretches or flips when multiplied by A, never rotating to a new direction. The corresponding eigenvalue lambda is the scalar factor: Av = lambda*v. In geometric terms, eigenvectors mark the "axes of action" of the transformation, and eigenvalues measure how strongly each axis is scaled. For a 2x2 matrix, the two eigenvectors define the natural coordinate frame in which the transformation is purely a scaling. For a 3x3 matrix there are up to three such axes. Eigenvalues and eigenvectors appear throughout applied mathematics: principal component analysis in statistics, vibration modes in structural engineering, Google's PageRank algorithm, quantum mechanics' energy levels, and stability analysis of differential equations all rely on them.
How to find eigenvalues: the characteristic polynomial
To find the eigenvalues of matrix A, form the characteristic equation det(A - lambda*I) = 0, where I is the identity matrix. Expanding the determinant gives a polynomial in lambda (degree 2 for a 2x2 matrix, degree 3 for a 3x3 matrix). The roots of this polynomial are the eigenvalues. For a 2x2 matrix the characteristic polynomial is lambda^2 - tr(A)*lambda + det(A) = 0, solved directly with the quadratic formula. For a 3x3 matrix you get a cubic polynomial, which this calculator solves using the trigonometric (three real roots) or Cardano (one real root) method. Once you have each eigenvalue lambda_i, you find the corresponding eigenvector by row-reducing the homogeneous system (A - lambda_i*I)v = 0 and extracting the null-space vector.
Trace, determinant and their connection to eigenvalues
The trace of a matrix (sum of its diagonal entries) always equals the sum of its eigenvalues. The determinant always equals the product of its eigenvalues. These two facts give quick sanity checks: if your computed eigenvalues do not add up to the trace or multiply to the determinant, there is an arithmetic error somewhere. The determinant also tells you about invertibility: if any eigenvalue is zero the determinant is zero and the matrix is singular, meaning it maps some nonzero vectors to zero and cannot be inverted.
Diagonalization: what it means and when it is possible
A matrix A is diagonalizable if you can write it as A = P*D*P^-1, where D is a diagonal matrix containing the eigenvalues and P is a matrix whose columns are the corresponding eigenvectors. This decomposition greatly simplifies repeated matrix multiplication (A^k = P*D^k*P^-1) and solving systems of differential equations. An n x n matrix with n distinct eigenvalues is always diagonalizable. If eigenvalues repeat, diagonalizability requires that the geometric multiplicity (dimension of the eigenspace) equals the algebraic multiplicity (the root's multiplicity in the characteristic polynomial) for each repeated eigenvalue. A matrix that fails this condition is called defective and cannot be fully diagonalized, though it does have a Jordan normal form.
Key matrix properties from eigenvalues
| Property | Formula | What it tells you |
|---|---|---|
| Trace | tr(A) = sum of eigenvalues | Sum of diagonal entries |
| Determinant | det(A) = product of eigenvalues | 0 means singular (not invertible) |
| Rank deficiency | Number of zero eigenvalues | Dimension of the null space |
| Spectral radius | max |lambda_i| | Controls iteration convergence |
| Positive definite | All eigenvalues > 0 | Used in optimization and stability |
| Symmetric matrix | All eigenvalues real, eigenvectors orthogonal | Spectral Theorem (PSD, PD, etc.) |
| Diagonalizable | n independent eigenvectors | A = PDP^-1 decomposition exists |
These relationships hold for any square matrix A with eigenvalues lambda1, lambda2, ..., lambdaN.
Frequently asked questions
What is the difference between algebraic and geometric multiplicity?
Algebraic multiplicity is how many times a particular eigenvalue appears as a root of the characteristic polynomial. Geometric multiplicity is the dimension of the eigenspace for that eigenvalue, which equals the number of linearly independent eigenvectors associated with it. Algebraic multiplicity is always greater than or equal to geometric multiplicity. If they are equal for every eigenvalue, the matrix is diagonalizable. If geometric multiplicity is strictly less than algebraic multiplicity for any eigenvalue, the matrix is defective.
Can eigenvalues be negative or complex?
Yes. Eigenvalues can be any real number including negative values and zero. They can also be complex numbers that appear as conjugate pairs when the matrix has real entries. This calculator reports only real eigenvalues; if the characteristic polynomial has no real roots (discriminant negative for a 2x2 case), it notes that eigenvalues are complex. Physically, a negative eigenvalue means the eigenvector direction is reversed (a reflection), while a complex pair corresponds to a rotation-and-scaling rather than a pure stretch.
Why does this calculator normalize eigenvectors to unit length?
Any non-zero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue, so the direction matters but the length is arbitrary. Normalizing to unit length (dividing by the vector's magnitude) gives a canonical form that is easy to compare and use in further calculations such as building the diagonalizing matrix P. If you need eigenvectors scaled differently, multiply the displayed unit vector by any non-zero scalar.
What does it mean if the determinant is zero?
A zero determinant means at least one eigenvalue is zero. The matrix is singular: it maps some non-zero input vector to the zero vector, so it is not invertible. In systems of equations terms, the homogeneous system Ax = 0 has non-trivial solutions. In applied contexts a zero eigenvalue signals that the transformation collapses one or more dimensions, which can indicate redundancy in a dataset (PCA) or a structural mechanism in engineering.
Does the order of eigenvalues matter?
Mathematically, no: the set of eigenvalues (the spectrum) is a property of the matrix regardless of ordering. This calculator lists them in descending order by value. What does matter is pairing each eigenvalue consistently with its own eigenvector: eigvec1 corresponds to lambda1, eigvec2 to lambda2, and so on. When you build the diagonalizing matrix P, the columns must be ordered to match the corresponding diagonal entries of D.