Eigenvector Calculator
Compute linearly independent eigenvectors (v), eigenspace nullity kernels, basis vectors, and verify directional invariance under linear matrix transformations.
Geometric Eigen-Axes on Cartesian Plane
Av = λvStep-by-Step Eigenspace Kernel Derivation: (A − λI)v = 0 Nullspace Basis Vectors
How to Calculate Matrix Eigenvectors
To calculate the eigenvectors of an n × n square matrix A, first determine its eigenvalues λ by solving det(A - λI) = 0. For each distinct eigenvalue λ, construct the characteristic matrix (A - λI) by subtracting λ from each diagonal entry. Set up the homogeneous linear matrix equation (A - λI)v = 0. Perform Gaussian elimination or row reduction to reduced row echelon form (RREF). Identify the free variables, express basic variables in terms of free parameters, and write the basis vectors spanning the eigenspace Null(A - λI). Normalize the basis vectors if an orthonormal coordinate system is needed.
Definition of Eigenvectors & The Nullspace Kernel
While eigenvalues represent scalar magnification factors, eigenvectors represent the fundamental axes of stability in a linear transformation. When a square matrix A transforms space, most arbitrary vectors are simultaneously rotated and stretched. An eigenvector v is an exceptional vector whose spatial span remains completely unaltered by the transformation:
The linear transformation A acts purely as a scalar multiplier along the 1D subspace spanned by v.
To convert this geometric requirement into an algorithmic procedure, we collect all vectors onto the left side of the equation and introduce the identity matrix I:
In abstract linear algebra, the solution set of all vectors v that are mapped to the zero vector by an operator M is called the nullspace or kernel of M, denoted Null(M). Therefore, every eigenvector corresponding to eigenvalue λ belongs to the nullspace of the shifted matrix (A - λI). If you need to calculate eigenvalues first, explore our Eigenvalue Calculator.
Finding Eigenspace Basis via Gaussian Elimination
Because λ was chosen specifically as a root of the characteristic determinant det(A - λI) = 0, the rows of (A - λI) are guaranteed to be linearly dependent. Consequently, Gaussian row elimination will always create at least one row of zeros:
For any non-trivial system of linear equations, you can also cross-verify your nullspace kernel using our dedicated System of Equations Calculator.
Eigenvector Normalization & Directional Rays
An essential property of linear equations is homogeneity: if v is an eigenvector of A, then any scalar multiple c · v (for c ≠ 0) is also an eigenvector:
Thus, an eigenvector does not represent a single point, but an entire invariant line (ray) passing through the origin. To standardize eigenvectors across scientific literature, machine learning libraries, and numerical algorithms, vectors are conventionally normalized to unit Euclidean length (||v|| = 1):
Normalized eigenvectors ensure numerical stability when assembling the modal transformation matrix P in diagonalization and matrix powers. Compute dot products and verify unit lengths using our Dot Product Calculator.
Geometric vs Algebraic Multiplicity & Defective Spaces
When an eigenvalue repeats as a root of the characteristic polynomial, the dimension of its eigenspace determines whether the matrix is diagonalizable:
Algebraic Multiplicity (AM)
The exponent of the linear factor (λ - λ₀) in the characteristic polynomial. Measures how many times λ₀ appears as a root.
Geometric Multiplicity (GM)
The dimension of the nullspace dim(Null(A - λ₀I)). Measures the number of linearly independent eigenvectors associated with λ₀.
It is an unbreakable mathematical theorem that 1 ≤ GM ≤ AM. If GM < AM, the matrix has an "eigenvector deficit" and is called defective. For example, the shear matrix [[2, 1], [0, 2]] has λ = 2 with AM = 2, but row reduction reveals only 1 linearly independent eigenvector (GM = 1).
Orthogonal Eigenbases & The Real Spectral Theorem
For real symmetric matrices (Aᵀ = A), the Spectral Theorem provides an extraordinary structural guarantee:
- Real Spectrum: All eigenvalues are guaranteed to be real numbers; complex conjugate eigenvalues cannot occur.
- Orthogonal Eigenspaces: Eigenvectors corresponding to distinct eigenvalues are mutually perpendicular: v₁ · v₂ = 0.
- Full Orthonormal Basis: Even if eigenvalues repeat (AM > 1), the geometric multiplicity always equals the algebraic multiplicity (GM = AM), so symmetric matrices are never defective!
This property enables orthogonal matrix diagonalization A = Q D Qᵀ, where Q is an orthogonal rotation matrix whose columns are normalized eigenvectors. To multiply and invert modal transformation matrices, explore our Matrix Multiplication Calculator.
Step-by-Step Worked Calculation Examples
Given matrix A = [[4, 1], [2, 3]] with eigenvalues λ₁ = 5 and λ₂ = 2, find both eigenvectors.
Find the eigenvectors of the horizontal shear matrix A = [[3, 2], [0, 3]].
Common Pitfalls & Diagnostic Table
| Mistake | Erroneous Practice | Correct Mathematical Rule |
|---|---|---|
| Zero Vector Confusion | Reporting v = [0, 0]ᵀ as an eigenvector | Eigenvectors must be strictly non-zero (v ≠ 0) by mathematical axiom. |
| Assuming Unique Vector | Thinking there is only one exact eigenvector answer | Any scalar multiple c·v is equally valid; they span the exact same invariant 1D subspace. |
| Row Reducing A Directly | Row reducing matrix A to find eigenvectors | Row operations alter eigenvectors! You must row reduce the shifted matrix (A - λI). |
| Defectiveness Blindness | Assuming repeated eigenvalues always give multiple vectors | When GM < AM, the matrix is defective and lacks a full basis of eigenvectors. |
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