SVD Calculator

- May 11, 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 factorisation 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

What is the SVD Calculator?

The SVD Calculator is a handy tool that helps users compute the Singular Value Decomposition (SVD) of a matrix. This process breaks down a matrix into three parts: U, Σ, and V*. It's a powerful technique used in various fields such as Statistics, image compression, and data analysis. With this calculator, anyone can grasp SVD without needing complex calculations by hand.

How Does the SVD Work?

When you input a matrix into the SVD Calculator, it performs a factorisation. The result is three matrices: U, which contains the left singular vectors; Σ, which holds the singular values on its diagonal; and V*, which features the right singular vectors. This breakdown is essential for understanding the structure of the original matrix and has practical applications across different domains.

Understanding the Key Components of SVD

  • Matrix U: An orthogonal matrix representing the left singular vectors.
  • Matrix Σ: A diagonal matrix with non-negative singular values.
  • Matrix V*: The conjugate transpose of an orthogonal matrix, showcasing right singular vectors.

Each of these components plays a significant role in understanding the properties and behaviour of the original matrix. By examining them, users can gain insights into the matrix’s rank, condition number, and Other key features.

Applications of the SVD Calculator

The SVD Calculator is useful in various fields. It can help with:

  • Reducing dimensions in data sets (like PCA).
  • Improving image compression for smaller file sizes without losing much quality.
  • Enhancing noise reduction in data analysis.
  • Providing recommendations in systems that suggest products or services.

These applications highlight the versatility of SVD, making it crucial for anyone working with data or matrices.

Number of Rows and Columns

Users can easily specify the dimensions of the matrix by simply choosing the number of rows and columns. The calculator allows for a maximum of 10 rows and 10 columns. This flexibility makes it suitable for a range of matrix sizes, whether for small data sets or more extensive operations.

Real-Time Results and Verification

Once the calculations are complete, the SVD Calculator presents the decomposition results clearly. Users can see the original matrix alongside the U, Σ, and V* matrices. The calculator also verifies the result by showing users that the original matrix can be reconstructed using the formula A = U·Σ·V*. This verification ensures that users can trust the calculations and comprehend the relationship between the components.

Customisation Options for Better Understanding

The SVD Calculator offers several display options. Users can choose how many decimal places to show, making it easier to read and understand the results. There's also an option to show explanations and highlight significant values. These customisation features cater to both beginners and more advanced users, providing clarity when analysing the results.

Embracing the Power of SVD

In summary, the SVD Calculator is an invaluable tool for anyone looking to understand matrix decomposition. Whether you're a student, a researcher, or simply curious about matrices, this calculator simplifies the process. With its user-friendly interface and robust features, you can explore the fascinating world of Singular Value Decomposition effortlessly.