Eigenvalues and Eigenvectors
Eigenvalues λ and eigenvectors v satisfy Av=λv — the matrix scales v without rotating it. For 2×2: λ²−tr(A)λ+det(A)=0. Sum of eigenvalues = trace; product = determinant. Essential for PageRank, PCA, vibration analysis.
Did our AI summary help? Let us know.
Eigenvectors define directions that the matrix only scales. Complex eigenvalues occur for rotation-like transformations. Diagonalizable matrices: A=PDP⁻¹ where D has eigenvalues.
Ready to run the numbers?
Why: Eigenvalues determine matrix behavior: stability of dynamical systems, principal directions in PCA, natural frequencies in vibration. Google PageRank uses the dominant eigenvector.
How: Solve det(A−λI)=0 for eigenvalues. For each λ, solve (A−λI)v=0 for eigenvectors. 2×2: λ=(tr±√(tr²−4·det))/2. Trace and determinant give quick checks.
Run the calculator when you are ready.
📐 Real-World Examples — Click to Load
Matrix Entries
Eigenvalue Magnitudes
Trace Decomposition (Eigenvalue Contribution)
📐 Calculation Breakdown
Eigenvectors
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
PageRank: dominant eigenvector (λ=1) of link matrix ranks pages.
PCA: eigenvectors of covariance matrix are principal components.
— ML
📋 Key Takeaways
- • Eigenvalues λ satisfy Av = λv — the matrix scales the eigenvector without rotating it
- • For 2×2: λ² - (a+d)λ + (ad-bc) = 0 where trace = a+d, determinant = ad-bc
- • Sum of eigenvalues = trace; product = determinant
- • Complex eigenvalues occur when discriminant < 0 (e.g., rotation matrices)
💡 Did You Know?
📖 How It Works
The eigenvalue equation Av = λv means: when you multiply the matrix A by an eigenvector v, you get a scalar multiple of v. The scalar λ is the eigenvalue. Geometrically, A only stretches or shrinks v; it does not rotate it.
To find eigenvalues: solve det(A - λI) = 0. For 2×2, this gives the quadratic λ² - tr(A)λ + det(A) = 0. Use the quadratic formula. For each eigenvalue, solve (A - λI)v = 0 to get eigenvectors.
🎯 Expert Tips
Symmetric Matrices
Real symmetric matrices have real eigenvalues and orthogonal eigenvectors. Essential for PCA.
Rotation Matrices
90° rotation [[0,-1],[1,0]] has eigenvalues ±i — complex eigenvalues indicate rotation.
Stability
For dynamical systems: |λ| < 1 means stable; |λ| > 1 means growing/unstable.
Trace & Determinant
Quick check: λ₁+λ₂=tr(A), λ₁λ₂=det(A). Use to verify your eigenvalues.
📊 Quick Reference
| Property | Formula |
|---|---|
| Trace | tr(A) = a₁₁ + a₂₂ |
| Determinant (2×2) | det(A) = a₁₁a₂₂ - a₁₂a₂₁ |
| Characteristic (2×2) | λ² - tr·λ + det = 0 |
| Eigenvalue sum | λ₁ + λ₂ = tr(A) |
| Eigenvalue product | λ₁ · λ₂ = det(A) |
❓ FAQ
What does Av = λv mean?
The eigenvector v, when multiplied by A, is only scaled by λ. The direction of v is unchanged (or reversed if λ < 0).
When do complex eigenvalues occur?
When the discriminant tr² - 4·det < 0. Common for rotation matrices. They come in conjugate pairs a±bi.
How is this used in data science?
PCA: eigenvectors of the covariance matrix are principal components; eigenvalues give variance explained.
What is the geometric interpretation?
Eigenvectors are directions that the matrix only stretches. Eigenvalues are the stretch factors.
Why does Google use eigenvalues?
PageRank finds the eigenvector for λ=1 of the link matrix — the steady-state distribution of "importance."
Can a matrix have no eigenvalues?
Over the reals, 2×2 can have 0 or 2 (real) eigenvalues. Over complex numbers, every n×n matrix has n eigenvalues.
📚 Official Sources
⚠️ Disclaimer: This calculator supports 2×2 matrices (full) and 3×3 (characteristic polynomial, numerical eigenvalues). For research or production, verify with MATLAB, NumPy, or similar. Complex eigenvector display is simplified for 2×2.
Related Calculators
Matrix Calculator
Add, subtract, scalar multiply, transpose, and invert 2×2 and 3×3 matrices. Step-by-step solutions, matrix properties (symmetric, diagonal), Bar and Doughnut...
MathematicsMatrix Determinant Calculator
Calculate determinant of 2×2 (ad-bc) and 3×3 (Sarrus rule, cofactor expansion) matrices. Singular matrix check, cofactor matrix, adjugate. Geometric interpretation (area/volume), Cramer's rule. Bar and Doughnut charts.
MathematicsAbsolute Value Equation Calculator
Solve absolute value equations and inequalities including |ax+b|=c, |ax+b|=|cx+d|, and |ax+b|≤c forms. Features solution verification, number line...
MathematicsAbsolute Value Inequalities Calculator
Solve absolute value inequalities |ax+b| < c, > c, ≤ c, ≥ c. Get solution intervals, interval notation, set builder notation, number line visualization...
MathematicsBessel Function Calculator
Calculate Bessel functions J_n(x), Y_n(x), I_n(x), and K_n(x) with series approximation. Supports first kind (J), second kind/Neumann (Y), and modified Bessel functions (I, K). Applications include vibrating drum modes, electromagnetic waveguides, heat conduction in cylinders, and Bessel filter design in signal processing. Uses the power series J_n(x) = Σ (-1)^m/(m!(m+n)!)·(x/2)^(2m+n). Includes convergence info, Bar chart comparing values at different x, Doughnut chart showing series term contributions, and educational content on Bessel's differential equation and cylindrical harmonics.
MathematicsBinomial Coefficient Calculator
Calculate binomial coefficients C(n,k) = n!/(k!(n-k)!) — the number of ways to choose k items from n without regard to order. Also known as n choose k or ⁿCₖ. Features Pascal's triangle row generation, factorial breakdown, Bar chart of full row C(n,0)..C(n,n), Doughnut chart showing k vs n−k. Applications: lottery (C(49,6)), poker hands (C(52,5)), committee selection, DNA combinations, binary strings. Educational content on combinatorics, binomial theorem (x+y)ⁿ expansion, and Pascal's identity.
Mathematics