1. What is Singular Value Decomposition (SVD)?

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 equation:

Where:

• \( A \) — Input matrix (m × n)
• \( U \) — Orthogonal matrix (m × m)
• \( \Sigma \) — Diagonal matrix (m × n) with singular values
• \( V \) — Orthogonal matrix (n × n)

Explanation: The SVD decomposes any matrix into two orthogonal matrices and a diagonal matrix containing the singular values.

3. Applications of SVD

Details: SVD has applications in signal processing, statistics, semantic analysis, image compression, and principal component analysis.

4. Using the Calculator

Tips: Enter your matrix with each row on a new line and elements separated by spaces or commas. The matrix must be rectangular (all rows same length).

5. Frequently Asked Questions (FAQ)

Q1: What are singular values?
A: Singular values are non-negative real numbers that indicate the magnitude of the matrix's action in different directions.

Q2: How is SVD different from eigendecomposition?
A: SVD works for any m × n matrix, while eigendecomposition only works for square matrices. Also, SVD always exists and is numerically stable.

Q3: What do the U and V matrices represent?
A: U contains the left singular vectors (output directions), V contains the right singular vectors (input directions).

Q4: Are there limitations to SVD?
A: For very large matrices, computation can be expensive. Also, interpretation of results requires understanding of linear algebra concepts.

Q5: What are some practical uses of SVD?
A: Image compression (JPEG), recommendation systems, noise reduction, and solving linear least squares problems.