1. What is Singular Value Decomposition?

Singular Value Decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any m×n matrix via an extension of the polar decomposition.

2. How Does the Calculator Work?

The calculator computes the SVD using the formula:

Where:

• \( A \) — Input matrix (m×n)
• \( U \) — Left singular vectors (m×m orthogonal matrix)
• \( \Sigma \) — Diagonal matrix of singular values (m×n)
• \( V \) — Right singular vectors (n×n orthogonal matrix)

Explanation: The SVD breaks down any matrix into rotation and scaling components, represented by the orthogonal matrices U and V, and the scaling factors in Σ.

3. Applications of SVD

Details: SVD has numerous applications including data compression, noise reduction, solving linear equations, computer vision, principal component analysis, and more.

4. Using the Calculator

Tips: Enter your matrix with each row on a new line and elements separated by spaces or commas. The calculator will compute and display the U, Σ, and V matrices.

5. Frequently Asked Questions (FAQ)

Q1: What types of matrices can be decomposed with SVD?
A: SVD works for any m×n matrix, whether square or rectangular, real or complex.

Q2: How accurate is the SVD computation?
A: The accuracy depends on the implementation, but modern algorithms provide excellent numerical stability.

Q3: What do the singular values represent?
A: Singular values indicate the importance of each corresponding pair of singular vectors in reconstructing the original matrix.

Q4: Can SVD be used for low-rank approximations?
A: Yes, by keeping only the largest k singular values, you can create a rank-k approximation of the original matrix.

Q5: How does SVD relate to PCA?
A: Principal Component Analysis (PCA) is essentially SVD performed on mean-centered data (or using the covariance matrix).