Eigenvalue and Eigenvector Calculator

- May 11, 2025
| |

Calculate the eigenvalues and eigenvectors of a square matrix. Eigenvalues and eigenvectors have important applications in linear transformations, differential equations, quantum mechanics, and many other fields of mathematics and physics.

Matrix Input

Display Options

About Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself. This scalar is called the eigenvalue.

Mathematical Definition

For a square matrix A, a non-zero vector v is an eigenvector if:

Av = λv

where λ is the eigenvalue associated with the eigenvector v.

Finding Eigenvalues and Eigenvectors
  1. Eigenvalues: Solve the characteristic equation det(A - λI) = 0
  2. Eigenvectors: For each eigenvalue λ, solve the homogeneous system (A - λI)v = 0
Applications of Eigenvalues and Eigenvectors
  • Differential Equations: Solving systems of linear differential equations
  • Principal Component Analysis: Dimensionality reduction in statistics and data science
  • Quantum Mechanics: Describing physical observables and quantum states
  • Engineering: Vibration analysis, structural stability, and control systems
  • Matrix Diagonalization: Simplifying matrix operations through transformation
  • Graph Theory: Spectral analysis of networks and connectivity
Properties
  • The determinant of a matrix equals the product of its eigenvalues
  • The trace of a matrix equals the sum of its eigenvalues
  • A matrix is diagonalizable if and only if there are n linearly independent eigenvectors for an n×n matrix
  • Symmetric matrices always have real eigenvalues and orthogonal eigenvectors
  • Similar matrices share the same eigenvalues

Understanding Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are key concepts in Linear Algebra. They help us understand how linear transformations work. When you have a square matrix, an eigenvector is a special vector that doesn't change direction when the matrix is applied to it. The eigenvalue is the factor by which the eigenvector is scaled. These concepts are widely used in various fields like Physics, engineering, and data analysis.

Introducing the Eigenvalue and Eigenvector Calculator

The Eigenvalue and Eigenvector Calculator simplifies finding eigenvalues and eigenvectors for square matrices. It's designed for students, teachers, and professionals needing quick and accurate results. You can easily input a matrix and check the calculations, including any steps taken during the process, which can help in understanding how results were achieved.

How to Input Your Matrix

Using the calculator is straightforward. You start by selecting the size of the matrix. You can choose from 2x2 up to 6x6 matrices. After selecting the size, you can enter the values for each element of the matrix. The calculator generates a clear input area for you to fill in, making the setup simple and user-friendly.

Customizing Your Results

The calculator allows you to adjust how results are displayed. You can choose the number of decimal places for precision and decide if you want steps shown in calculations. There’s also an option to normalise eigenvectors and show complex results. This flexibility helps you tailor the output to your needs.

Viewing Eigenvalue and Eigenvector Results

Once the matrix is entered, the calculator provides results for both eigenvalues and eigenvectors. The results are clearly presented, showing the original matrix alongside its eigenvalues and corresponding eigenvectors. This layout makes it easy to compare the original data with the calculated values.

Verification of Results

To ensure the results are correct, the calculator includes a verification section. Here, it checks the equation Av = λv to confirm that the eigenvalues and eigenvectors satisfy this core relationship. This verification adds confidence in the accuracy of the calculations and enhances learning.

Applications of Eigenvalues and Eigenvectors

  • Differential Equations: Useful in solving linear systems.
  • Data Science: Help in dimensionality reduction through techniques like Principal Component Analysis.
  • Quantum Mechanics: Essential for understanding physical properties.
  • Engineering: Important in analysing vibrations and stability.

Learning More About Eigenvalues and Eigenvectors

The concepts of eigenvalues and eigenvectors are foundational in many areas of Mathematics and engineering. Understanding how to calculate and interpret these values can help in various applications, from scientific research to practical engineering challenges. With the provided calculator, you can practice and gain confidence in this essential topic.