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.

2. How Does SVD Work?

The SVD of an m × n matrix A is a factorization of the form:

Where:

• \( U \) — m × m orthogonal matrix (left singular vectors)
• \( \Sigma \) — m × n diagonal matrix with non-negative singular values
• \( V^T \) — n × n orthogonal matrix (right singular vectors, transposed)

Explanation: The singular values in Σ are the square roots of the eigenvalues of AᵀA, and the columns of U and V are the eigenvectors of AAᵀ and AᵀA respectively.

3. Applications of SVD

Details: SVD has numerous applications including data compression, noise reduction, pseudoinverse calculation, principal component analysis (PCA), and solving linear least squares problems.

4. Using the Calculator

Tips: Enter your matrix in the format "a,b,c;d,e,f" where semicolons separate rows and commas separate columns. The calculator will compute the SVD components U, Σ, and V.

5. Frequently Asked Questions (FAQ)

Q1: What are singular values?
A: Singular values are the diagonal entries of Σ and represent the "importance" or "energy" of each corresponding dimension in the transformed space.

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

Q3: What does orthogonality of U and V mean?
A: Their columns are orthonormal vectors, meaning UᵀU = I and VᵀV = I (identity matrices).

Q4: How can SVD be used for dimensionality reduction?
A: By keeping only the largest singular values (truncated SVD), you can approximate the original matrix with fewer dimensions.

Q5: Is the SVD unique?
A: The singular values are uniquely determined, but U and V are not unique - their columns can change sign.