Eigenvalues and Eigenvectors Calculator
Eigenvalues and eigenvectors are the backbone of linear algebra — they power Google's PageRank, compress images, reveal vibration modes in bridges, and drive machine learning. This calculator finds eigenvalues and eigenvectors for 2×2 and 3×3 matrices instantly, showing the full characteristic polynomial, quadratic/cubic solution steps, and null-space eigenvectors. Choose your matrix size below and enter your values.
Step 2 — Solve for λ: roots of the characteristic polynomial are eigenvalues
Step 3 — Find eigenvectors: for each λ, solve (A − λI)v = 0
Enter your matrix in the tool below — all three steps are shown automatically.
What Are Eigenvalues and Eigenvectors?
Every square matrix A represents a linear transformation — it takes a vector and maps it to another vector, possibly rotating, reflecting, scaling, or shearing it. For most vectors, the direction changes after the transformation. But for special vectors called eigenvectors, the direction is preserved — the matrix just scales them by a constant factor. That constant is the eigenvalue.
Formally, a non-zero vector v is an eigenvector of matrix A with eigenvalue λ if:
This equation says: "applying A to v gives the same result as multiplying v by the scalar λ." If λ = 3, the transformation triples the length of v. If λ = −1, it flips v. If λ = 0, the vector collapses to zero — meaning v is in the null space of A.
- An n×n matrix has exactly n eigenvalues (counting multiplicity, over complex numbers)
- Eigenvectors are never the zero vector — by definition
- Scalar multiples of an eigenvector are also eigenvectors for the same eigenvalue
- Different eigenvalues always have linearly independent eigenvectors
- The set of all eigenvectors for a given λ (plus the zero vector) is called the eigenspace
The Characteristic Equation
To find eigenvalues, rearrange Av = λv to (A − λI)v = 0. For this system to have a non-zero solution, the matrix (A − λI) must be singular — its determinant must be zero. This gives the characteristic equation:
Expanding this determinant produces the characteristic polynomial p(λ). Its degree equals the matrix size:
- 2×2 matrix: p(λ) = λ² − tr(A)·λ + det(A) — a quadratic, solved with the quadratic formula
- 3×3 matrix: p(λ) = −λ³ + tr(A)·λ² − (sum of 2×2 principal minors)·λ + det(A) — a cubic
- n×n matrix: degree-n polynomial with up to n real roots
det(A − λI) = (a−λ)(d−λ) − bc = λ² − (a+d)λ + (ad−bc)
→ λ = [(a+d) ± √((a+d)² − 4(ad−bc))] / 2
→ λ = [tr(A) ± √(tr(A)² − 4·det(A))] / 2
How to Find Eigenvectors
Once you have an eigenvalue λ, substitute it into (A − λI)v = 0 and solve for the vector v using row reduction (Gaussian elimination). The solution is a free-variable family — you pick any non-zero representative as the eigenvector.
2. Row-reduce (A − λI) to row echelon form.
3. Express the system of equations from the reduced matrix.
4. Identify the free variable(s) — set them to 1 (or any non-zero value).
5. Back-substitute to find the other components.
6. Write the eigenvector as [v₁, v₂, …]ᵀ.
For a 2×2 matrix, (A − λI) has rank 1 when λ is an eigenvalue, so there is always one free variable and one eigenvector direction. For repeated eigenvalues (algebraic multiplicity > 1), the eigenspace may have dimension 1 or 2 — this determines whether the matrix is diagonalizable.
2×2 Step-by-Step Example
Find the eigenvalues and eigenvectors of:
Step 1 — Characteristic polynomial:
det(A − λI) = (4−λ)(3−λ) − (1)(2)
= λ² − 7λ + 12 − 2 = λ² − 7λ + 10 = 0
Step 2 — Solve the quadratic:
λ = [7 ± √(49−40)] / 2 = [7 ± √9] / 2 = [7 ± 3] / 2
λ₁ = 5, λ₂ = 2
Step 3 — Eigenvector for λ₁ = 5:
(A − 5I) = [[-1, 1], [2, -2]]
Row reduce → [[-1, 1], [0, 0]]
Equation: −v₁ + v₂ = 0 → v₁ = v₂ → eigenvector: [1, 1]ᵀ
Step 4 — Eigenvector for λ₂ = 2:
(A − 2I) = [[2, 1], [2, 1]]
Row reduce → [[2, 1], [0, 0]]
Equation: 2v₁ + v₂ = 0 → v₂ = −2v₁ → eigenvector: [1, −2]ᵀ
Verification: A·[1,1]ᵀ = [5,5]ᵀ = 5·[1,1]ᵀ ✓ A·[1,−2]ᵀ = [2,−4]ᵀ = 2·[1,−2]ᵀ ✓
3×3 Step-by-Step Example
For 3×3 matrices, the characteristic polynomial is a cubic. If the matrix is upper or lower triangular, the eigenvalues are simply the diagonal entries — no cubic needed.
Since A is upper triangular, eigenvalues are the diagonal entries:
λ₁ = 2, λ₂ = 3, λ₃ = 4
Eigenvector for λ₁ = 2:
(A − 2I) = [[0,5,3],[0,1,1],[0,0,2]]
Row reduce → free variable: v₁ (x-component), pivot rows fix v₂=0, v₃=0
Eigenvector: [1, 0, 0]ᵀ
General 3×3 note: For non-triangular matrices, the cubic characteristic polynomial
p(λ) = −λ³ + tr(A)λ² − (M₁₁+M₂₂+M₃₃)λ + det(A) = 0
where M₁₁, M₂₂, M₃₃ are the 2×2 principal cofactors. Use the calculator above for exact numerical roots.
Properties of Eigenvalues
Eigenvalues encode key information about the matrix itself. These relationships let you sanity-check your results quickly:
- Sum of eigenvalues = trace of A → λ₁ + λ₂ + … + λₙ = tr(A) = a₁₁ + a₂₂ + … + aₙₙ
- Product of eigenvalues = determinant of A → λ₁ · λ₂ · … · λₙ = det(A)
- If λ is an eigenvalue of A, then λᵏ is an eigenvalue of Aᵏ (matrix power)
- If A is invertible, 1/λ is an eigenvalue of A⁻¹
- Singular matrix ↔ at least one eigenvalue is zero (det = 0 ↔ 0 is an eigenvalue)
- Real symmetric matrices always have real eigenvalues and orthogonal eigenvectors
- Eigenvalues of a diagonal matrix are its diagonal entries
- Similar matrices (B = P⁻¹AP) share the same eigenvalues
tr(A) = 4+3 = 7 → eigenvalues: 5+2 = 7 ✓
det(A) = 4·3−1·2 = 10 → eigenvalues: 5×2 = 10 ✓
Special Matrix Types and Their Eigenvalues
Recognizing the matrix type can immediately tell you properties of its eigenvalues, saving significant calculation effort:
| Matrix Type | Eigenvalue Property | Example |
|---|---|---|
| Symmetric (Aᵀ = A) | All eigenvalues real; eigenvectors orthogonal | [[3,1],[1,3]] → λ = 4, 2 |
| Diagonal | Eigenvalues = diagonal entries | diag(2,5,−1) → λ = 2, 5, −1 |
| Upper/Lower Triangular | Eigenvalues = diagonal entries | [[2,7],[0,3]] → λ = 2, 3 |
| Identity matrix I | λ = 1 (multiplicity n) | 3×3 I → λ = 1, 1, 1 |
| Zero matrix | λ = 0 (multiplicity n) | 2×2 zeros → λ = 0, 0 |
| Rotation matrix (2D) | Complex eigenvalues e^{±iθ} | [[0,−1],[1,0]] → λ = ±i |
| Orthogonal (AᵀA = I) | |λ| = 1 (magnitude 1) | Reflection: λ = 1 or −1 |
| Positive definite | All eigenvalues > 0 | Covariance matrices |
| Idempotent (A² = A) | λ = 0 or λ = 1 only | Projection matrices |
| Nilpotent (Aᵏ = 0) | All eigenvalues = 0 | [[0,1],[0,0]] → λ = 0, 0 |
Quick Reference: Formulas at a Glance
| Quantity | Formula |
|---|---|
| Characteristic equation | det(A − λI) = 0 |
| 2×2 characteristic polynomial | λ² − tr(A)λ + det(A) = 0 |
| 2×2 eigenvalues | λ = [tr(A) ± √(tr(A)² − 4·det(A))] / 2 |
| 3×3 trace | tr(A) = a₁₁ + a₂₂ + a₃₃ |
| 3×3 determinant | Cofactor expansion along first row |
| Find eigenvectors | Row-reduce (A − λI); null space = eigenvectors |
| Eigenvalue sum check | λ₁ + λ₂ + … = tr(A) |
| Eigenvalue product check | λ₁ × λ₂ × … = det(A) |
| Matrix power eigenvalue | Eigenvalues of Aᵏ are λᵢᵏ |
| Inverse eigenvalue | Eigenvalues of A⁻¹ are 1/λᵢ |
| Discriminant (2×2) | Δ = tr(A)² − 4·det(A) |
| Δ > 0 | Two distinct real eigenvalues |
| Δ = 0 | One repeated real eigenvalue |
| Δ < 0 | Two complex conjugate eigenvalues |
Real-Life Applications of Eigenvalues and Eigenvectors
- Google PageRank — The web is modeled as a matrix where each entry represents a link between pages. The dominant eigenvector of this matrix gives the "rank" of every page on the internet. Google's original algorithm is literally the power iteration method on a billion-node matrix.
- Principal Component Analysis (PCA) — In machine learning, data dimensionality is reduced by finding the eigenvectors of the covariance matrix (principal components). The eigenvectors with the largest eigenvalues capture the most variance in the data.
- Structural engineering — Eigenvalues of a stiffness matrix give the natural frequencies at which a bridge, building, or aircraft wing vibrates. Engineers design structures so these frequencies avoid resonance with external forces (wind, traffic, earthquakes).
- Quantum mechanics — In quantum physics, observable quantities (energy, momentum, spin) are eigenvalues of Hermitian operators. The famous Schrödinger equation is an eigenvalue problem where energy levels are eigenvalues of the Hamiltonian.
- Image compression and SVD — Singular Value Decomposition, which powers JPEG compression and Netflix recommendations, is intimately tied to eigenvalues of AᵀA and AAᵀ.
- Stability of differential equations — For a system ẋ = Ax, the eigenvalues of A determine stability: all eigenvalues with negative real part → system decays to zero (stable); any with positive real part → unstable (exponential blow-up).
- Markov chains and population dynamics — Long-run behavior of a Markov chain (like animal population models or economic models) is governed by the dominant eigenvector of the transition matrix.
- Computer graphics and 3D rotations — Eigenvalues/vectors of inertia tensors determine how 3D objects rotate, used in game physics engines and CAD software.
- Network analysis — In social networks and epidemiology, the largest eigenvalue of the adjacency matrix determines the epidemic threshold — below it, a disease dies out; above it, it spreads.
Common Mistakes to Avoid
(A − λI) means you subtract λ from a₁₁, a₂₂, a₃₃ — NOT from every entry of A. Off-diagonal entries remain unchanged.
If your row reduction leaves a row like [0, 0, 0], that is correct — it means you have a free variable. It does NOT mean there is an error. Never set a zero row equal to "no solution."
By definition, eigenvectors must be non-zero. v = [0, 0]ᵀ trivially satisfies Av = λv for any λ, but it is excluded from the definition precisely because it is uninformative. Always pick a non-zero vector from the null space.
Algebraic multiplicity is how many times λ is a root of the characteristic polynomial. Geometric multiplicity is the dimension of the eigenspace. A matrix is diagonalizable if and only if they are equal for every eigenvalue. If geometric < algebraic for any eigenvalue, the matrix is defective (not diagonalizable).
Both give valid characteristic polynomials, but they differ by a sign factor (−1)ⁿ. For even-sized matrices they are identical; for odd-sized they differ in sign. Your eigenvalues are the same either way — only the polynomial's leading coefficient changes. Be consistent with whichever convention your textbook uses.
Glossary
- Eigenvalue (λ)
- A scalar such that Av = λv has a non-zero solution v. It tells you how much the transformation stretches or compresses an eigenvector.
- Eigenvector (v)
- A non-zero vector whose direction is unchanged (only scaled) by the linear transformation A. Associated with a specific eigenvalue.
- Characteristic Polynomial
- p(λ) = det(A − λI). The polynomial whose roots are the eigenvalues of A. Degree n for an n×n matrix.
- Characteristic Equation
- The equation det(A − λI) = 0. Solving it gives the eigenvalues.
- Eigenspace
- The set of all eigenvectors for a given eigenvalue λ, together with the zero vector. It forms a subspace of ℝⁿ.
- Trace (tr A)
- The sum of diagonal entries of A. Equals the sum of all eigenvalues.
- Determinant (det A)
- A scalar computed from a square matrix. Equals the product of all eigenvalues. Zero iff the matrix is singular.
- Algebraic Multiplicity
- The number of times an eigenvalue appears as a root of the characteristic polynomial.
- Geometric Multiplicity
- The dimension of the eigenspace for a given eigenvalue; equals the number of linearly independent eigenvectors for that eigenvalue.
- Diagonalizable Matrix
- A matrix A that can be written as A = PDP⁻¹, where D is diagonal and P's columns are eigenvectors. Requires enough linearly independent eigenvectors (algebraic = geometric multiplicity for all λ).
- Power Iteration
- A numerical algorithm that repeatedly multiplies a starting vector by A (then normalizes) to converge to the dominant eigenvector and eigenvalue.
- Dominant Eigenvalue
- The eigenvalue with the largest absolute value. The one found by power iteration.
Frequently Asked Questions
Q: What are eigenvalues and eigenvectors?
A: An eigenvector is a non-zero vector v such that Av = λv — the matrix A only scales it by the eigenvalue λ without changing its direction. Geometrically, eigenvectors are the "invariant directions" of a linear transformation.
Q: How do you find eigenvalues of a matrix?
A: Solve the characteristic equation det(A − λI) = 0. For a 2×2 matrix this gives a quadratic, solved with the quadratic formula. For a 3×3 matrix it gives a cubic, typically solved numerically or by inspection for special matrices (triangular, symmetric, etc.).
Q: How do you find eigenvectors from eigenvalues?
A: Substitute each eigenvalue λ back into (A − λI)v = 0. Row-reduce the resulting matrix to echelon form, identify the free variable(s), set them to 1, and back-substitute to find the remaining components. The resulting non-zero vector is an eigenvector.
Q: What is the characteristic polynomial?
A: p(λ) = det(A − λI). For a 2×2 matrix A = [[a,b],[c,d]], it is λ² − (a+d)λ + (ad−bc). Setting it to zero gives the characteristic equation whose roots are eigenvalues.
Q: Can a matrix have complex eigenvalues?
A: Yes. Rotation matrices (e.g., 90° rotation) have complex eigenvalues e^{±iθ} because they have no real invariant directions. Complex eigenvalues always come in conjugate pairs when A has real entries. Symmetric matrices are guaranteed to have only real eigenvalues.
Q: What is the difference between eigenvalue and eigenvector?
A: An eigenvalue is a scalar (number) measuring the scale factor. An eigenvector is a direction (vector) that the transformation leaves invariant. Every eigenvalue has at least one eigenvector; multiple eigenvectors can share the same eigenvalue (forming an eigenspace).
Q: What are eigenvalues used for in real life?
A: Google PageRank uses the dominant eigenvector of the web's link matrix. PCA in machine learning uses eigenvectors of the covariance matrix. Structural engineers use eigenvalues to find resonant frequencies. Physicists use them for quantum energy levels. Image compression (SVD) relies on them. They are also central to Markov chains, stability analysis, and network theory.