Singular Value Decomposition (SVD) Calculator
What is Singular Value Decomposition (SVD)?
Singular Value Decomposition (SVD) is a fundamental technique in linear algebra and matrix theory. It is a factorization method that decomposes a matrix into three separate matrices, revealing important properties and structures of the original matrix. SVD has numerous applications in various fields, including data compression, dimensionality reduction, signal processing, and machine learning.
The SVD of a matrix A is expressed as:A = U * Σ * V^T
Where:
- A is the original m x n matrix
- U is an m x m orthogonal matrix (left singular vectors)
- Σ (Sigma) is an m x n diagonal matrix with non-negative real numbers (singular values)
- V^T is the transpose of an n x n orthogonal matrix V (right singular vectors)
The singular values in Σ are typically arranged in descending order, representing the importance of each “component” in the decomposition. This property makes SVD particularly useful for dimensionality reduction and data compression tasks.
How to use the SVD Calculator?
1. Matrix Input
- Start by specifying the dimensions of your matrix using the “Rows” and “Columns” input fields. You can enter any size from 1×1 up to 10×10.
- Click the “Generate Matrix” button to create an empty matrix with the specified dimensions.
- Fill in the matrix cells with your numerical values. You can use integers or decimal numbers.
2. Calculation
- Once you’ve entered all the values in your matrix, click the “Calculate SVD” button.
- The tool will perform the Singular Value Decomposition using the math.js library.
3. Results
- The calculator will display the three component matrices of the SVD:
a. U matrix (left singular vectors)
b. Σ (Sigma) matrix (diagonal matrix of singular values)
c. V^T matrix (transpose of right singular vectors) - Each matrix is presented in a clear, tabular format with values rounded to four decimal places for readability.
4. Verification
- To confirm the accuracy of the decomposition, the calculator multiplies U * Σ * V^T and displays the result.
- It also calculates and shows the maximum difference between the original matrix and the reconstructed matrix, which should be very close to zero if the decomposition is accurate.
This SVD Calculator is an excellent tool for students learning linear algebra, researchers working with large datasets, or professionals in fields like computer vision and signal processing. It provides a quick and easy way to perform SVD on matrices, visualize the results, and verify the decomposition’s accuracy.
By offering both the computational results and a brief explanation, it serves as both a calculation tool and a learning aid for understanding the principles of Singular Value Decomposition.