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. It decomposes a matrix A into three matrices: U, Σ, and V^T.

2. How Does SVD Work?

The SVD of an m × n matrix A is given by:

Where:

• \( U \) — m × m orthogonal matrix (left singular vectors)
• \( \Sigma \) — m × n diagonal matrix (singular values)
• \( V \) — n × n orthogonal matrix (right singular vectors)

Explanation: The singular values in Σ are always non-negative and arranged in descending order. The columns of U and V are orthonormal bases for the column and row spaces of A.

3. Applications of SVD

Details: SVD has numerous applications including data compression, principal component analysis (PCA), solving linear systems, image processing, and recommendation systems.

4. Using the Calculator

Tips: Enter your matrix with elements separated by spaces or commas, and rows separated by new lines. The calculator will compute the U, Σ, and V matrices.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between SVD and eigendecomposition?
A: SVD works for any rectangular matrix, while eigendecomposition only works for square matrices. SVD always exists and is numerically stable.

Q2: How are singular values related to eigenvalues?
A: The singular values of A are the square roots of the eigenvalues of A^TA or AA^T.

Q3: What does the rank of a matrix have to do with SVD?
A: The rank equals the number of non-zero singular values in Σ.

Q4: Can SVD be used for low-rank approximations?
A: Yes, by keeping only the largest k singular values (and corresponding vectors), you get the best rank-k approximation of the matrix.

Q5: Is SVD unique?
A: The singular values are unique, but the singular vectors may not be (they can change sign or rotate within subspaces corresponding to identical singular values).