Emathhelp SVD Calculator

Singular Value Decomposition (SVD):

\[ A = U \Sigma V^T \]

where:

\( U \) is m × m orthogonal matrix
\( \Sigma \) is m × n diagonal matrix with non-negative singular values
\( V \) is n × n orthogonal matrix

Matrix U:

Matrix Σ (Singular Values):

Matrix V:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 equation:

\[ A = U \Sigma V^T \]

Where:

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

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 rows separated by newlines 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 are singular values?
A: Singular values are non-negative real numbers that provide important geometric and algebraic information about the matrix.

Q2: How accurate is this calculator?
A: The accuracy depends on the implementation of the SVD algorithm. For critical applications, verify with specialized numerical software.

Q3: Can I use complex numbers?
A: This calculator currently only supports real-valued matrices.

Q4: What's the largest matrix size I can compute?
A: Performance depends on your server configuration, but very large matrices may time out or exhaust memory.

Q5: How are the singular values ordered?
A: Typically in descending order along the diagonal of Σ.