Eigenvalue and Eigenvector Calculator

Matrix A =
Identity Matrix I =
c =
Number matrix (Z=c×I)
|A| =
The trace of the matrix A =
Singular Matrix(A - c×I) =
|A - c×I| =
Eigenvalue (c1) =
Eigenvalue (c2) =
The value of c1 in the eigenvector x1 =
The value of c1 in the eigenvector x2 =
The value of c2 in the eigenvector x1 =
The value of c2 in the eigenvector x2 =

An Eigenvalue and Eigenvector Calculator is a tool (either online, in software like MATLAB or Python, or in calculators like TI-89) used to compute the eigenvalues and eigenvectors of a given square matrix.

What are Eigenvalues and Eigenvectors?

  • Eigenvalues (λ) are special scalars associated with a square matrix A. They satisfy the equation: Av=λv where v is a nonzero vector (eigenvector).
  • Eigenvectors are the corresponding vectors that remain in the same direction after transformation by A, only scaled by the eigenvalue.

Why are Eigenvalues and Eigenvectors Important?

  1. Engineering & Physics: Used in stability analysis, vibrations, quantum mechanics, etc.
  2. Machine Learning: Principal Component Analysis (PCA) for dimensionality reduction.
  3. Computer Graphics: Used in transformations like rotations and scaling.
  4. Differential Equations: Solving systems of linear differential equations.
  5. Graph Theory: Used to analyze networks.

How Does an Eigenvalue and Eigenvector Calculator Work?

  1. Input: A square matrix A.
  2. Compute Eigenvalues:
    • Solve the characteristic equation: det⁡(A−λI)=0
    • Find values of λ\lambda λ (roots of the determinant equation).
  3. Compute Eigenvectors:
    • Solve (A − λI)v = 0 for each λ.

When to Use It?

  • When dealing with large matrices (solving by hand is impractical).
  • In scientific computing, machine learning, engineering, etc.
  • When performing PCA, stability analysis, or quantum mechanics calculations.