Svd Calculator Emath

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 (Orthogonal):

Matrix Σ (Diagonal):

Matrix V (Orthogonal):

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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 commas and elements separated by spaces. The calculator will compute the U, Σ, and V matrices.

5. Frequently Asked Questions (FAQ)

Q1: What are singular values?
A: Singular values are the square roots of the eigenvalues of \( A^TA \). They appear in descending order on the diagonal of Σ.

Q2: How accurate is this calculator?
A: The calculator uses numerical methods to compute the SVD with high precision.

Q3: Can I use this for complex matrices?
A: This calculator is designed for real matrices. Complex matrices require a different approach.

Q4: What if my matrix is not full rank?
A: The SVD works for any matrix, regardless of rank. The number of non-zero singular values equals the rank.

Q5: How can I use SVD for matrix approximation?
A: By keeping only the largest k singular values (and corresponding vectors), you can create a rank-k approximation of the original matrix.