Input Matrix (A)
Define a square matrix to compute its eigenvalues and eigenvectors.
Eigensystem Results
Transformation Visualizer
How to Use the Calculator
Get your complete eigensystem analysis in two steps.
Define Your Matrix
Use the stepper to set the size of your square matrix (A). Then, input the values for each element of the matrix.
Analyze the Results
The calculator instantly provides the eigenvalues (λ) and their corresponding eigenvectors (v), along with a transformation visualizer for 2x2 matrices.
What are Eigenvectors & Eigenvalues?
The special vectors that reveal the true nature of a linear transformation.
In linear algebra, when a matrix (representing a linear transformation) is multiplied by a vector, the vector is typically stretched, shrunk, and rotated. However, for any given matrix A, there exist special non-zero vectors called eigenvectors.
When the matrix A acts on an eigenvector v, the resulting vector points in the exact same direction as the original. It is only scaled (stretched or shrunk) by a specific factor. This scaling factor is called the eigenvalue (λ).
This fundamental relationship is captured by the equation:
Finding these pairs is crucial as they represent the fundamental directions or "axes" of the transformation. Our eigenvectors of a matrix calculator automates this discovery process.
An Advanced Analysis Lab
Features designed for clarity and deep geometric insight.
Transformation Visualizer
For 2x2 matrices, our live plot shows how the basis vectors (i and j) are transformed, and highlights the eigenvectors which remain unchanged in direction.
High-Precision Results
Get accurate eigenvalues and normalized eigenvectors, clearly presented in pairs. The calculator handles both real and complex results seamlessly.
Dynamic Sizing
Easily set the dimensions of your square matrix from 2x2 up to 5x5 using the intuitive stepper controls. The input grid adapts instantly.
Applications of Eigensystems
Eigendecomposition is a cornerstone of science and data analysis.
Principal Component Analysis (PCA)
In data science, eigenvectors of the covariance matrix represent the principal components (directions of highest variance), crucial for dimensionality reduction.
Quantum Mechanics
In quantum physics, observables (like energy or momentum) are represented by matrices. Their eigenvalues are the possible measured values of that observable.
Google's PageRank
The original PageRank algorithm models the web as a massive matrix. The eigenvector corresponding to the largest eigenvalue gives the "rank" of each page.
