Spectral Decomposition Calculator

Spectral decomposition refers to the process of breaking down a square matrix into its eigenvalues and eigenvectors. These eigenvalues and eigenvectors represent the intrinsic properties of the matrix and can be used for various mathematical and physical analyses, such as solving systems of linear equations, understanding the behavior of dynamic systems, and more.

About the Author: Created by Fotios Angelakis, MSc in Mechanical Engineering, with 5+ years of experience in data analytics and energy engineering. Learn more about the author's qualifications and experience.

Choose Decomposition Type:




How the Spectral Decomposition Calculator Works

This calculator uses eigenvalue decomposition to break a square matrix into useful components. It is commonly applied in engineering, physics, and data analysis.

Input Parameters

  • Matrix Size – Select the dimensions (e.g., 2×2, 3×3).
  • Decomposition Type – Choose to compute eigenvalues, eigenvectors, or the full decomposition.
  • Numerical Tolerance – Sets the precision of the calculation (e.g., 1e-6).

General Formula

A = V · Λ · V-1

  • A → original matrix
  • V → matrix of eigenvectors
  • Λ → diagonal matrix of eigenvalues
  • V-1 → inverse of V

Example 1: 2×2 Matrix

A = [2, 3]
    [3, 4]

After applying spectral decomposition:

  • Eigenvalues: [6.162278, -0.162278]
  • Eigenvectors matrix (V):
[0.584710, -0.811242]
[0.811242,  0.584710]

The reconstructed matrix using A = V · Λ · V⁻¹ returns the original values. This confirms the accuracy of the decomposition.

Example 2: 3×3 Matrix

Consider another case:

A = [ 4, 1, 0 ]
    [ 1, 3, 1 ]
    [ 0, 1, 2 ]

Approximate eigenvalues:

  • λ₁ = 5.236
  • λ₂ = 2.414
  • λ₃ = 1.350

Eigenvectors form matrix V:

V ≈ 
[  0.82,  0.40, -0.41 ]
[  0.49, -0.82,  0.30 ]
[  0.29,  0.40,  0.86 ]

Diagonal matrix (Λ):

[ 5.236, 0,     0     ]
[ 0,     2.414, 0     ]
[ 0,     0,     1.350 ]

Applications of this type of matrix include:

  • Structural mechanics
  • Heat transfer simulations
  • Vibration analysis
  • Principal Component Analysis (PCA)

FAQs

What does the decomposition show?

It reveals key properties about a matrix, such as stability, dominant directions, and energy distribution.

What is the purpose of the reconstructed matrix?

It verifies that the decomposition is valid by showing that multiplying V · Λ · V⁻¹ returns the original matrix.

Do I need a square matrix?

Yes. Spectral decomposition is only defined for square matrices.

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