Linear Algebra SVD Calculator

Singular Value Decomposition (SVD):

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

Where:

\( A \) - m × n matrix to decompose
\( U \) - m × m orthogonal matrix
\( \Sigma \) - m × n diagonal matrix with non-negative singular values
\( V \) - 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 fundamental matrix factorization technique in linear algebra that decomposes any m×n matrix into three component matrices: U, Σ, and Vᵀ. It's widely used in signal processing, statistics, and machine learning.

2. How Does SVD Work?

The SVD decomposes a matrix A into:

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

Where:

U is an m×m orthogonal matrix (left singular vectors)
Σ is an m×n diagonal matrix with non-negative singular values
V is an n×n orthogonal matrix (right singular vectors)

Key Properties: The singular values in Σ are always non-negative and typically ordered from largest to smallest. The number of non-zero singular values equals the rank of the matrix.

3. Applications of SVD

Common Uses: Principal Component Analysis (PCA), data compression, noise reduction, solving linear systems, pseudoinverse computation, and recommendation systems.

4. Using the Calculator

Instructions: Enter your matrix using commas or spaces to separate elements and semicolons to separate rows. For example, "1,2,3;4,5,6" represents a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between SVD and eigenvalue decomposition?
A: SVD works for any matrix (including rectangular ones), while eigenvalue decomposition 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 AᵀA (or AAᵀ).

Q3: What do the U and V matrices represent?
A: U's columns are eigenvectors of AAᵀ, V's columns are eigenvectors of AᵀA. They represent orthogonal bases for the column and row spaces.

Q4: When is SVD unique?
A: SVD is unique up to sign flips of singular vectors when singular values are distinct. Repeated singular values have corresponding subspaces that can be rotated.

Q5: How is SVD used in data science?
A: In PCA for dimensionality reduction, in recommender systems (collaborative filtering), and for low-rank approximations that reduce noise in data.