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Eigenvalues (concept) Calculator

Characteristic equation of a matrix.

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Overview

An eigenvalue is a scalar factor that represents how a corresponding eigenvector is scaled during a linear transformation. In the fundamental relationship A v = λ v, the eigenvalue λ determines whether the vector v is stretched, compressed, or flipped without changing its span or orientation in space.

Symbols

Variables

Top-left element of the 2x2 matrix. = Matrix Element (1,1), Top-right element of the 2x2 matrix. = Matrix Element (1,2), Bottom-left element of the 2x2 matrix. = Matrix Element (2,1), Bottom-right element of the 2x2 matrix. = Matrix Element (2,2), Specifies the type of eigenvalue-related calculation to perform. = Calculation Type

Matrix Element (1,1)
Matrix Element (1,2)
Matrix Element (2,1)
Matrix Element (2,2)
Calculation Type
Result

Apply it well

When To Use

When to use: Use eigenvalues when diagonalizing a matrix to simplify computation, analyzing the stability of a linear system of differential equations, or identifying the principal components in a large dataset. They are essential for understanding systems where a transformation preserves the direction of certain input vectors.

Why it matters: Eigenvalues define the natural frequencies of vibration in structural engineering, preventing mechanical failure due to resonance. They also underpin the PageRank algorithm used by search engines to rank the importance of web pages and are used in quantum mechanics to represent observable physical quantities like energy levels.

Avoid these traps

Common Mistakes

  • Arithmetic errors in determinant.
  • Forgetting identity matrix I.

One free problem

Practice Problem

Find the larger eigenvalue λ for a 2×2 matrix where the first row is [6, 2] and the second row is [2, 3].

Matrix Element (1,1)6
Matrix Element (1,2)2
Matrix Element (2,1)2
Matrix Element (2,2)3
Calculation TypelargerEigenvalue largerEigenvalue

Solve for:

Hint: Solve the characteristic equation det(A - λI) = 0, which results in (6-λ)(3-λ) - 4 = 0.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Eigenvalues and eigenvectors
  2. Introduction to Linear Algebra by Gilbert Strang
  3. Strang, Linear Algebra and Its Applications
  4. Griffiths, Introduction to Quantum Mechanics
  5. Lay, Linear Algebra and Its Applications
  6. Linear Algebra and Its Applications by Gilbert Strang
  7. Eigenvalues and eigenvectors (Wikipedia article)
  8. Standard curriculum — Linear Algebra