Eigenvalue & Eigenvector Calculator

Find eigenvalues and eigenvectors of 2x2 and 3x3 matrices with the characteristic polynomial and step-by-step solutions.

Enter Matrix

Matrix A (2x2):

Results

Eigenvalues
5, 2
of the given matrix
Trace (sum of diagonal) 7
Determinant 10
Characteristic Polynomial --

Step-by-Step Solution

det(A - λI) = 0

Understanding Eigenvalues and Eigenvectors

An eigenvector of a square matrix A is a nonzero vector v such that when A is multiplied by v, the result is a scalar multiple of v. The scalar multiplier is called the eigenvalue. In equation form: Av = λv, where λ (lambda) is the eigenvalue and v is the corresponding eigenvector.

Geometrically, an eigenvector points in a direction that is preserved (only scaled, not rotated) by the linear transformation represented by the matrix. The eigenvalue tells you how much the vector is scaled.

How to Find Eigenvalues and Eigenvectors

Step 1: Characteristic Equation

Set up det(A - λI) = 0, where I is the identity matrix.

det(A - λI) = 0

Step 2: Expand Determinant

Expand to get the characteristic polynomial in λ.

λ² - tr(A)λ + det(A) = 0 (2x2)

Step 3: Solve for λ

Find roots of the characteristic polynomial. These are the eigenvalues.

Use quadratic formula or factoring

Step 4: Find Eigenvectors

For each eigenvalue λ, solve (A - λI)v = 0.

(A - λI)v = 0

2x2 Shortcut

For a 2x2 matrix, eigenvalues come from a simple quadratic.

λ = (tr ± sqrt(tr² - 4det)) / 2

Properties

Sum of eigenvalues = trace; product of eigenvalues = determinant.

Σλ = tr(A), Πλ = det(A)

Applications of Eigenvalues and Eigenvectors

  • Principal Component Analysis (PCA): Eigenvectors of the covariance matrix identify the directions of maximum variance in data.
  • Google PageRank: The PageRank algorithm finds the dominant eigenvector of the web link matrix.
  • Quantum Mechanics: Observable quantities correspond to eigenvalues of Hermitian operators.
  • Vibration Analysis: Natural frequencies of structures are eigenvalues of the stiffness-mass matrix system.
  • Differential Equations: Solutions to linear ODE systems involve matrix eigenvalues and eigenvectors.
  • Stability Analysis: Eigenvalues determine if a system is stable (all eigenvalues have negative real parts).

Special Types of Matrices

  • Symmetric matrices: Always have real eigenvalues and orthogonal eigenvectors.
  • Diagonal matrices: Eigenvalues are the diagonal entries.
  • Triangular matrices: Eigenvalues are the diagonal entries.
  • Orthogonal matrices: Eigenvalues have absolute value 1.
  • Positive definite: All eigenvalues are positive.

Tips and Common Mistakes

  • Eigenvectors are not unique; any scalar multiple of an eigenvector is also an eigenvector.
  • The zero vector is never an eigenvector by definition.
  • Complex eigenvalues always come in conjugate pairs for real matrices.
  • A matrix is singular (non-invertible) if and only if 0 is an eigenvalue.