Eigenvalue Equation Calculator
Solve the eigenvector relation A·v = λ·v for any unknown in the equation — eigenvalues λ, or any matrix entry given a λ — with step-by-step derivations, a live characteristic-polynomial chart, a complex-plane diagram of the roots, and full eigenvector verification with Av = λv.
Eigenvectors are the directions the matrix does not rotate
Applied to a general vector, a matrix rotates and scales it — it produces a new vector pointing somewhere else. But for a special set of directions, the matrix only scales the vector; it does not turn it. Those directions are the eigenvectors, and the scaling factors are the eigenvalues. The equation A·v = λ·v says exactly this: applying A to v is the same as multiplying v by the scalar λ. In geometrical terms, the matrix "stretches" along its eigenvector directions and "spins" everything else.
The characteristic polynomial reduces it to a quadratic
Rearranging A·v = λ·v as (A − λI)·v = 0, this system has a nonzero solution v if and only if det(A − λI) = 0. For 2×2 matrices this determinant expands to a quadratic in λ: λ² − tr(A)·λ + det(A) = 0, where tr(A) = a + d is the trace and det(A) = ad − bc is the determinant. The eigenvalues are the roots. Their sum is always the trace; their product is always the determinant.
The discriminant tells you real vs. complex
The discriminant Δ = tr² − 4·det decides the character of the roots. If Δ > 0, there are two distinct real eigenvalues — the matrix stretches along two different real directions. If Δ = 0, there is one repeated eigenvalue — the matrix has a single eigen-direction (it is defective unless it is a scalar multiple of the identity). If Δ < 0, the eigenvalues are a complex conjugate pair and the matrix has no real eigenvectors — geometrically, it combines a rotation with a scaling.
Eigenvalues drive stability, vibration, and PCA
Eigenvalues are the single most useful summary of a linear operator. In differential equations, the sign of the real part of λ determines whether a fixed point is stable or unstable. In physics and engineering, the eigenvalues of a stiffness or mass matrix are the natural frequencies of a vibrating system. In principal component analysis (PCA), the eigenvectors of the covariance matrix are the principal directions of variance, and the eigenvalues measure how much variance lies along each direction. Whenever a system can be described by a square matrix, its eigenvalues are the key to understanding its behaviour.
How to Use This Calculator
The eigenvalue equation links a 2×2 matrix, its eigenvalues, and its eigenvectors. Pick which quantity you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick λ to compute the eigenvalues from a given 2×2 matrix, or pick any of a, b, c, or d to solve for one matrix entry given the eigenvalue λ and the other three entries.
2. Use the matrix presets
The top-right selector loads classic matrices: diagonal, symmetric, rotation, shear, singular, and random — each with a different eigenvalue structure to explore.
3. Watch the matrix display update live
The 2×2 display above the entry grid shows the current matrix. When you solve for an entry, its cell is highlighted in amber.
4. Read the result and verdict
The result card shows both eigenvalues (real or complex), plus the trace, determinant, and discriminant, with a colour-coded verdict: two real distinct, one repeated, or complex conjugate pair.
5. Verify with the eigenvector panel
The eigenvector card computes v for each eigenvalue and verifies Av = λv numerically, showing both sides of the equation side by side.
6. Switch to Advanced for depth
Advanced mode adds sensitivity rows, a companion metrics panel with trace/det/eigenvalue relations, and a deeper geometric and algebraic analysis.
What This Tool Offers
Built around the eigenvalue equation A·v = λ·v — the fundamental relation that identifies the invariant directions of a linear transformation.
Five-way solving
Solve for the eigenvalues λ, or for any matrix entry a, b, c, d given the eigenvalue and the other three entries.
Eigenvector verification
The calculator computes an eigenvector for each real eigenvalue and verifies Av = λv numerically, showing the two vectors side by side.
Characteristic polynomial chart
The polynomial f(λ) = λ² − tr·λ + det is plotted, with the real roots highlighted and the vertex marked at λ = tr/2.
Complex plane diagram
Both eigenvalues are plotted on the complex plane, showing whether they are real, repeated, or a conjugate pair.
Eigenvalue classification
Each result is classified as two real distinct, one repeated, or complex conjugate, with the geometric interpretation that follows.
Reference tables
Eigenvalues of common matrices, key identities (sum = trace, product = det), and the underlying variables.
Reference Tables & Conversions
Eigenvalues of common matrices, key identities, and the underlying variables.
Special matrices and their eigenvalues
Common 2×2 matrices with their eigenvalue structure.
| Matrix | Eigenvalues | Type | Geometric meaning |
|---|---|---|---|
| [[1,0],[0,1]] | 1, 1 | Repeated | Identity — every direction is an eigenvector |
| [[2,0],[0,3]] | 2, 3 | Real distinct | Diagonal — stretches axes independently |
| [[3,1],[1,3]] | 4, 2 | Real distinct | Symmetric — orthogonal eigenvectors |
| [[0,−1],[1,0]] | ±i | Complex conjugate | Pure rotation by 90° — no real eigenvector |
| [[1,1],[0,1]] | 1, 1 | Defective | Shear — only one eigenvector direction |
| [[0,1],[1,0]] | 1, −1 | Real distinct | Swap — reflection across y = x |
| [[1,2],[2,4]] | 0, 5 | Real (one zero) | Rank-1 — collapses one direction to zero |
| [[2,4],[1,2]] | 0, 4 | Real (one zero) | Singular — same structure as above |
Key Eigenvalue Identities
These relations hold for every 2×2 matrix and are the reason the characteristic polynomial can be written down without computing the determinant.
| Identity | Formula | Meaning |
|---|---|---|
| Sum of eigenvalues | λ₁ + λ₂ = tr(A) = a + d | The trace is always the sum of eigenvalues |
| Product of eigenvalues | λ₁ · λ₂ = det(A) = ad − bc | The determinant is always the product of eigenvalues |
| Characteristic polynomial | λ² − tr(A)·λ + det(A) = 0 | Roots of this quadratic are the eigenvalues |
| Discriminant | Δ = tr² − 4·det = (λ₁ − λ₂)² | Δ ≥ 0 → real, Δ = 0 → repeated, Δ < 0 → complex |
| Eigenvalue zero | λ = 0 ⟺ det = 0 | Matrix is singular exactly when 0 is an eigenvalue |
| Determinant of powers | det(Aⁿ) = λ₁ⁿ · λ₂ⁿ | Powers scale eigenvalues accordingly |
Variable Reference
| Symbol | Meaning | Range / Type | Notes |
|---|---|---|---|
| a, b, c, d | Matrix entries | Any real number | Row 1 and row 2 of the 2×2 matrix |
| λ | Eigenvalue | Real or complex | Root of the characteristic polynomial |
| v | Eigenvector | Nonzero 2-vector | Satisfies (A − λI)·v = 0 |
| tr(A) | Trace | a + d | Sum of eigenvalues |
| det(A) | Determinant | ad − bc | Product of eigenvalues |
| Δ | Discriminant | tr² − 4·det | Determines real vs. complex eigenvalues |
What Does the Eigenvalue Equation A·v = λ·v Measure?
The eigenvalue equation A·v = λ·v is one of the most powerful relations in all of mathematics. It says that applying a matrix A to a special vector v produces the same result as simply multiplying that vector by a scalar λ. The vector v is called an eigenvector (from German "eigen" meaning "own" or "characteristic"), and the scalar λ is its eigenvalue. The equation captures, in one compact form, the idea of an invariant direction: a direction along which the matrix only stretches or shrinks, but never rotates.
To find eigenvalues, you rewrite the equation as (A − λI)·v = 0, where I is the identity matrix. For this homogeneous system to have a nonzero solution v, the matrix (A − λI) must be singular, so its determinant must be zero. For a 2×2 matrix, this determinant expands into a quadratic in λ: λ² − tr(A)·λ + det(A) = 0. This is the characteristic polynomial, and its roots are the eigenvalues. The two most useful facts about this quadratic are that the sum of its roots is always the trace of A and the product of its roots is always the determinant of A.
Eigenvalues determine the qualitative behaviour of almost every physical and computational system that can be modelled as a matrix. They are the frequencies of a vibrating structure, the principal variances in PCA, the rates of growth or decay in a linear dynamical system, the stability exponents of a fixed point in a differential equation, and the possible measurement outcomes in quantum mechanics. Learning to solve the eigenvalue equation for 2×2 matrices is the first step toward understanding all of these applications.
The Eigenvalue Formula
The defining relation is:
Rearranging to isolate λ:
For a nonzero solution, the matrix must be singular, so its determinant is zero:
Expanding the determinant for a 2×2 matrix gives the characteristic polynomial:
The eigenvalues are the roots:
λ₁ = (tr(A) + √Δ) / 2
λ₂ = (tr(A) − √Δ) / 2
When Δ ≥ 0, both roots are real. When Δ < 0, the roots are the complex conjugate pair (tr/2) ± i·√(−Δ)/2.
An eigenvector for a real eigenvalue λ can be read directly off the matrix:
Eigenvalue Variables
| Symbol | Meaning | Notes |
|---|---|---|
| a, b, c, d | Matrix entries | a = (1,1), b = (1,2), c = (2,1), d = (2,2) |
| λ₁, λ₂ | Eigenvalues | Roots of λ² − tr·λ + det = 0 |
| v | Eigenvector | Nonzero vector satisfying (A − λI)·v = 0 |
| tr(A) | Trace | = a + d = λ₁ + λ₂ |
| det(A) | Determinant | = ad − bc = λ₁ · λ₂ |
| Δ | Discriminant | = tr² − 4·det = (λ₁ − λ₂)² |
How Each Entry Affects the Eigenvalues
The two eigenvalues depend on the matrix only through the trace and the determinant. So changing any entry affects the eigenvalues only insofar as it affects a + d or ad − bc. Increasing both a and d (the diagonal entries) increases the trace and typically shifts the eigenvalues upward; increasing b or c (the off-diagonal entries) decreases the determinant and can push the eigenvalues toward complex conjugates. The most sensitive direction is usually the one that changes the discriminant Δ = tr² − 4·det, because that is what determines whether the roots are real or complex.
A crucial geometric property is that the sum of the two eigenvalues is always exactly the trace, and their product is always exactly the determinant. These are called the trace–determinant identities, and they hold for every 2×2 matrix, real or complex. They provide a fast sanity check: if you compute eigenvalues and their sum does not equal a + d, or their product does not equal ad − bc, something is wrong.
When the discriminant Δ is exactly zero, the two eigenvalues coincide — the matrix has a repeated eigenvalue equal to tr/2. In that case the matrix is either a scalar multiple of the identity (in which case every vector is an eigenvector) or it is defective (in which case only a one-dimensional subspace of eigenvectors exists, even though the eigenvalue has algebraic multiplicity 2). This distinction matters in applications: defective matrices cannot be diagonalised, and their behaviour under iteration or exponentiation is qualitatively different from that of diagonalisable ones.
Eigenvalue Calculation Examples
tr(A) = 3 + 3 = 6
det(A) = 3·3 − 1·1 = 8
λ² − 6λ + 8 = 0 ⟹ (λ − 4)(λ − 2) = 0
λ₁ = 4, λ₂ = 2
tr(A) = 0, det(A) = 0·0 − (−1)·1 = 1
λ² + 1 = 0 ⟹ λ = ± i (complex)
tr(A) = 2, det(A) = 1
λ² − 2λ + 1 = 0 ⟹ (λ − 1)² = 0
λ = 1 (repeated, defective)
tr(A) = 7, det(A) = 4·3 − 1·2 = 10
λ² − 7λ + 10 = 0 ⟹ (λ − 5)(λ − 2) = 0
λ₁ = 5, λ₂ = 2
(a − λ)(d − λ) = bc ⟹ (a − 4)(−1) = 1 ⟹ a − 4 = −1
a = 3
Frequently Asked Questions
An eigenvalue λ of a square matrix A is a scalar such that there exists a nonzero vector v (the eigenvector) satisfying A·v = λ·v. Geometrically, λ is the factor by which the matrix stretches or shrinks the vector v without rotating it. The set of all eigenvalues is called the spectrum of A, and it captures the essential behaviour of the linear transformation. For a 2×2 matrix there are exactly two eigenvalues, counting multiplicity, and they are the roots of λ² − tr(A)·λ + det(A) = 0.
Compute the trace tr = a + d and the determinant det = ad − bc. Then solve the quadratic λ² − tr·λ + det = 0 using the quadratic formula: λ = (tr ± √(tr² − 4·det)) / 2. The quantity under the square root, Δ = tr² − 4·det, is the discriminant. If Δ > 0 you get two distinct real eigenvalues; if Δ = 0 you get one repeated eigenvalue equal to tr/2; if Δ < 0 you get a complex conjugate pair. That is the entire procedure for 2×2 matrices, and it takes only a few seconds to do by hand.
The eigenvalue λ is a number — the factor by which the matrix scales a particular direction. The eigenvector v is a direction — the actual vector that gets scaled by that factor. Every eigenvalue has a corresponding eigenspace (the set of all eigenvectors for that λ, plus the zero vector). The eigenvalue tells you "how much" and the eigenvector tells you "which direction". In applications, the eigenvalue determines stability, growth rate, or frequency; the eigenvector determines the mode shape or the principal direction.
A complex eigenvalue means the matrix has no real eigenvector direction — geometrically, the transformation combines a rotation with a scaling. The complex conjugate pair λ = σ ± iω corresponds to a rotation by angle arctan(ω/σ) with a scaling factor √(σ² + ω²). Rotation matrices like [[0,−1],[1,0]] have eigenvalues ±i, and their "eigenvectors" live in the complex vector space ℂ². In physics, complex eigenvalues typically indicate oscillatory behaviour (like a vibrating string or an AC circuit) rather than pure exponential growth or decay.
This follows from Vieta's formulas applied to the characteristic polynomial λ² − tr·λ + det = 0. Vieta's formulas say that for a quadratic aλ² + bλ + c = 0, the sum of the roots is −b/a and the product is c/a. Here a = 1, b = −tr, c = det, so the sum is tr and the product is det. These identities hold for every 2×2 matrix, real or complex, and they generalise: for an n×n matrix, the sum of all eigenvalues equals the trace, the product equals the determinant, and the intermediate coefficients of the characteristic polynomial are the elementary symmetric functions of the eigenvalues.
A defective matrix is a matrix with a repeated eigenvalue that has fewer independent eigenvectors than its algebraic multiplicity would suggest. The canonical 2×2 example is the shear matrix [[1,1],[0,1]], which has a repeated eigenvalue λ = 1 but only a one-dimensional eigenspace (spanned by (1,0)). Defective matrices cannot be diagonalised — they cannot be written as P·D·P⁻¹ with D diagonal. Instead, they have a Jordan normal form that includes an off-diagonal 1 in a 2×2 Jordan block. The distinction between diagonalisable and defective is important in applications: defective matrices behave qualitatively differently under iteration and exponentiation, often producing polynomial growth factors (like t·e^t) alongside the exponential factors.
Everywhere that linear algebra appears. In differential equations, eigenvalues determine stability of fixed points — negative real parts mean stable, positive mean unstable, imaginary parts mean oscillatory. In mechanical engineering, they are the natural frequencies of a vibrating structure; a bridge or building is designed so that its eigenvalues avoid expected excitation frequencies. In PCA, the covariance matrix eigenvalues rank principal directions by variance explained. In quantum mechanics, observable quantities are Hermitian operators whose eigenvalues are the possible measurement outcomes. In machine learning, eigenvalues underpin algorithms like spectral clustering, PageRank, and gradient descent convergence analysis. Even Google's original PageRank algorithm works by finding the dominant eigenvector of the web link matrix.
Yes, and it happens exactly when the determinant is zero. Recall λ₁ · λ₂ = det(A). If det(A) = 0, then at least one eigenvalue must be zero. A zero eigenvalue means the matrix collapses some nonzero direction to zero — there is a nonzero vector v with A·v = 0. That vector is in the null space of A, and the matrix is singular (not invertible). Geometrically, the transformation squashes the plane onto a line or point. Singular matrices have zero eigenvalues; invertible matrices have only nonzero eigenvalues.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| A | Square matrix | |
| v | Eigenvector | |
| λ | Eigenvalue |
What it means
The eigenvalue equation Av = λv is central to linear algebra and its applications. Here, A is a square matrix, v is a non‑zero vector (eigenvector), and λ is a scalar (eigenvalue). Geometrically, the matrix A transforms the vector v by stretching or compressing it by a factor λ, without changing its direction (or reversing it if λ is negative). Eigenvalues and eigenvectors are used to analyse linear systems: they determine stability (if eigenvalues have negative real parts), natural frequencies in vibration analysis (via the characteristic equation), and principal components in data analysis (PCA). In quantum mechanics, they represent observable quantities (energy levels). In engineering, they are used to solve differential equations and to design control systems. Finding eigenvalues involves solving the characteristic polynomial det(A−λI)=0. Understanding the eigenvalue equation is fundamental for advanced mathematics, physics, and engineering.
Worked example
Eigenvalue Equation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Matrix A | [[2,0],[0,3]] |
| Parameter | Value |
|---|---|
| Matrix A | [[3,1],[1,3]] |
Common mistakes
- Non‑zero vector: The eigenvector v must be non‑zero – the equation A v = λ v holds for non‑trivial solutions.
- Characteristic equation: Solve det(A − λI) = 0 to find eigenvalues – not det(A) = λ.
- Scaling: Eigenvectors are defined up to a scalar multiple – any non‑zero scalar multiple is also valid.
- Complex eigenvalues: May occur for real matrices – they come in conjugate pairs.
- Applications: Used in stability analysis, quantum mechanics, and many engineering problems.
Applications
The eigenvalue equation, Av = λv, states that a matrix A transforms a vector v only by scaling it by a scalar λ (the eigenvalue). This equation is central to many fields: in structural engineering, eigenvalues determine natural frequencies and mode shapes; in quantum mechanics, they represent energy levels; in machine learning, principal component analysis uses eigenvalues for dimensionality reduction. Engineers use eigenvalue analysis for stability assessments, vibration analysis, and buckling loads. In data science, eigenvalues help in covariance analysis and signal processing. Solving the eigenvalue problem is also essential for solving systems of differential equations. This simple equation unlocks a wealth of information about the system, making it a fundamental tool in applied mathematics and engineering.
- Vibration and modal analysis of mechanical structures
- Stability analysis of dynamical systems and control systems
- Principal component analysis and data reduction in statistics
- Quantum mechanics – solving the Schrödinger equation
- Solving coupled differential equations and system dynamics
Related calculators · Linear Algebra
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.