SVD Calculator

Category: Linear Algebra
- April 16, 2025
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Calculate the Singular Value Decomposition (SVD) of a matrix. SVD decomposes a matrix into three components: U, Σ, and V*. This decomposition has numerous applications in statistics, signal processing, image compression, and more.

Matrix Input

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About Singular Value Decomposition

Singular Value Decomposition (SVD) is a factorization technique that decomposes a matrix A into three matrices: U, Σ, and V*. It has wide applications in data science, signal processing, and linear algebra.

Mathematical Definition

For an m×n matrix A, the SVD is:

A = UΣV*

where:

  • U is an m×m orthogonal matrix (U*U = I)
  • Σ is an m×n diagonal matrix with non-negative singular values
  • V* is the conjugate transpose of an n×n orthogonal matrix V (V*V = I)
Applications of SVD
  • Dimensionality Reduction: Principal Component Analysis (PCA)
  • Image Compression: Reducing storage while maintaining image quality
  • Data Analysis: Noise reduction and feature extraction
  • Recommender Systems: Collaborative filtering for recommendations
  • Pseudoinverse Calculation: Finding solutions to ill-conditioned systems
  • Signal Processing: Filtering and denoising
Properties of SVD
  • Singular values are unique and arranged in descending order
  • Number of non-zero singular values equals the rank of the matrix
  • The condition number is the ratio of largest to smallest singular value
  • SVD can be used to compute the pseudoinverse via A⁺ = VΣ⁺U*
  • SVD is numerically stable for computation

Singular Value Decomposition Formula:

A = UΣV*

Where:

  • U = matrix of left singular vectors
  • Σ = diagonal matrix of singular values
  • V* = transpose of the matrix of right singular vectors

What Is the SVD Calculator?

The Singular Value Decomposition (SVD) Calculator is a web-based tool that breaks down any matrix into three fundamental components: U, Σ (Sigma), and V*. This process is widely used in linear algebra for applications such as data compression, signal filtering, and solving systems of equations.

With the SVD Calculator, you can quickly analyze the structure of a matrix, understand its dimensional properties, and compute key values like the condition number, rank, and pseudoinverse.

Why Use This Calculator?

This tool helps you:

  • Decompose a matrix into U, Σ, and V* for deeper analysis
  • Understand the behavior of your matrix through its singular values
  • Explore properties like the matrix rank, condition number, and Frobenius norm
  • Verify the accuracy of the decomposition
  • Compute the Moore-Penrose pseudoinverse instantly

If you're working with image data, statistics, or solving linear systems, this calculator gives you the breakdown you need. It complements other tools like the LU Decomposition Calculator, Pseudoinverse Calculator, and Matrix Inverse Calculator by focusing specifically on singular value analysis.

How to Use the SVD Calculator

Follow these simple steps:

  1. Select the number of rows and columns for your matrix.
  2. Click “Create Matrix” to generate input fields.
  3. Enter your matrix values into the fields.
  4. Choose display options such as decimal precision and whether to show step-by-step explanations.
  5. Click “Calculate SVD” to process the matrix.
  6. Review the resulting U, Σ, and V* matrices, as well as derived values like the pseudoinverse and condition number.

You can also load a sample matrix using the “Load Example” button to see how it works instantly.

Benefits of Using SVD

SVD is used across many fields and has practical benefits:

  • Data Compression: Reduce data size without losing essential features
  • Noise Reduction: Eliminate minor components in signals or datasets
  • Machine Learning: Improve algorithm performance via dimensionality reduction
  • Matrix Solutions: Compute the pseudoinverse when a normal inverse doesn't exist
  • Image Processing: Efficiently compress and reconstruct images

This tool is especially useful when paired with matrix tools such as the QR Factorization Calculator, Gauss-Jordan Elimination Calculator, and Diagonalize Matrix Calculator.

Frequently Asked Questions (FAQ)

  • What size matrix can I input?
    You can enter matrices from 1×1 up to 10×10.
  • What if my matrix has very small values?
    The calculator highlights near-zero values to help you focus on significant results.
  • Can I see how the SVD is calculated?
    Yes! Enable the “Show explanation” option to view a step-by-step breakdown of the decomposition.
  • Is this tool useful for learning?
    Definitely. The visual breakdown and calculation steps make it a great learning aid for students and educators.
  • How is this different from other matrix tools?
    While the Matrix Inverse Calculator and LU matrix factorization focus on direct solutions to linear systems, SVD offers more insight into the structure and stability of the matrix.

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