Eigen Decomposition Visualizer Calculator
Free Eigen decomposition visualizer Calculator for fractions. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
A = PDP^(-1) where det(A - lambda*I) = 0
Where A is the input matrix, P is the matrix of eigenvectors, D is the diagonal matrix of eigenvalues, and lambda represents each eigenvalue found by solving the characteristic equation det(A - lambda*I) = 0.
Worked Examples
Example 1: Symmetric Matrix Decomposition
Problem:Find the eigenvalues and eigenvectors of the symmetric matrix A = [[3, 1], [1, 3]].
Solution:Characteristic equation: lambda^2 - 6*lambda + 8 = 0\nDiscriminant = 36 - 32 = 4\nlambda1 = (6 + 2) / 2 = 4\nlambda2 = (6 - 2) / 2 = 2\nFor lambda1 = 4: (3-4)v1 + v2 = 0 => v = [1, 1]/sqrt(2)\nFor lambda2 = 2: (3-2)v1 + v2 = 0 => v = [-1, 1]/sqrt(2)\nVerification: trace = 3+3 = 6 = 4+2, det = 9-1 = 8 = 4*2
Result:Eigenvalues: 4 and 2 | Eigenvectors: [0.707, 0.707] and [-0.707, 0.707]
Example 2: Non-Symmetric Matrix Analysis
Problem:Decompose the matrix A = [[2, 1], [0, 3]] and verify the results.
Solution:Characteristic equation: (2-lambda)(3-lambda) = 0\nlambda1 = 3, lambda2 = 2\nFor lambda1 = 3: (2-3)v1 + v2 = 0 => v = [1, 1]/sqrt(2)\nFor lambda2 = 2: (2-2)v1 + v2 = 0 => v = [1, 0]\nVerification: A*[1,1] = [3,3] = 3*[1,1]\nA*[1,0] = [2,0] = 2*[1,0]
Result:Eigenvalues: 3 and 2 | Matrix is diagonalizable with condition number 1.5
Frequently Asked Questions
What is eigen decomposition and why is it important?
Eigen decomposition (also called eigendecomposition or spectral decomposition) is a factorization of a square matrix into a set of eigenvalues and eigenvectors. It decomposes a matrix A into the form A = PDP^(-1), where P is the matrix of eigenvectors and D is a diagonal matrix of eigenvalues. This decomposition is fundamental in linear algebra because it reveals the intrinsic geometric properties of the transformation that the matrix represents. It is widely used in principal component analysis, quantum mechanics, vibration analysis, and many other scientific and engineering applications where understanding the fundamental modes of a system is critical.
How is eigen decomposition used in Principal Component Analysis?
In Principal Component Analysis (PCA), eigen decomposition is applied to the covariance matrix of a dataset to find the directions of maximum variance. The eigenvectors of the covariance matrix become the principal components, and the corresponding eigenvalues indicate how much variance each component explains. The eigenvector with the largest eigenvalue points in the direction of greatest data spread. By selecting only the top k eigenvectors (those with the largest eigenvalues), you can reduce the dimensionality of your data while preserving the most important patterns. This technique is used extensively in data science, image compression, facial recognition, and genomics for dimensionality reduction.
Can every square matrix be eigen decomposed?
Not every square matrix can be eigen decomposed in the traditional sense. Defective matrices, which lack a full set of linearly independent eigenvectors, cannot be diagonalized. For example, the matrix [[1,1],[0,1]] has a repeated eigenvalue of 1 but only one linearly independent eigenvector, making it non-diagonalizable. However, every square matrix can be decomposed using the Jordan normal form, which generalizes eigendecomposition by allowing Jordan blocks for defective eigenvalues. Additionally, every matrix has a Schur decomposition (A = QTQ*) where T is upper triangular and Q is unitary, which always exists and is useful for numerical computation of eigenvalues.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy