1  Introduction to SVD

2 Introduction to Singular Value Decomposition (SVD)

2.1 What is SVD?

Singular Value Decomposition (SVD) is a fundamental matrix factorization technique that expresses a matrix as a product of three matrices:

\[ A = U \Sigma V^T, \quad A \in \mathbb{R}^{n \times p} \]

where:

  • \(U \in \mathbb{R}^{n \times n}\) is the matrix of left singular vectors,
  • \(\Sigma \in \mathbb{R}^{n \times p}\) is a diagonal matrix containing the singular values, and
  • \(V \in \mathbb{R}^{p \times p}\) is the matrix of right singular vectors.

This decomposition plays an important role in many areas of data analysis, including dimensionality reduction, signal processing, and more.

2.2 Intuition Behind SVD

The SVD provides a geometric interpretation of matrices. Each matrix can be viewed as performing a transformation of rotation, scaling, and again rotation. Here’s a breakdown:

  • The matrix \(V\) performs an initial rotation,
  • The matrix \(\Sigma\) scales along the principal directions (the singular values dictate how much scaling occurs),
  • The matrix \(U\) performs a final rotation.

By decomposing \(A\) into these three matrices, SVD reveals the key components and structure of the matrix.

2.3 Key Concepts

2.3.1 Singular Values

The singular values \(\sigma_i\) of a matrix are the entries of the diagonal matrix \(\Sigma\). These values are non-negative and are typically arranged in descending order:

\[ \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{\min(n, p)} \geq 0 \]

2.3.2 Matrix Norms and SVD

SVD plays a key role in minimizing matrix norms. Specifically, SVD provides the best low-rank approximation for a matrix, in terms of the Frobenius norm or the spectral norm.

Let \(A\) be any matrix, and let \(A_k\) be the matrix obtained by keeping only the top \(k\) singular values. Then, \(A_k\) minimizes the Frobenius norm:

\[ \| A - A_k \|_F \]

over all rank-\(k\) matrices.

2.3.3 Existence and Uniqueness

For any matrix \(A \in \mathbb{R}^{n \times p}\), the SVD always exists. While the singular values are unique, the singular vectors \(U\) and \(V\) may not be unique if some singular values are repeated.

2.4 Applications of SVD

SVD has a wide range of applications, including:

  • Low-Rank Approximation: Finding a rank-\(k\) matrix that approximates the original matrix.
  • Principal Component Analysis (PCA): A dimensionality reduction technique that uses SVD to identify the most important directions in data.
  • Latent Semantic Analysis (LSA): Used in text mining and natural language processing.
  • Image Compression: Compressing images by retaining only the most significant singular values.

Each of these applications will be explored in detail in subsequent chapters.

2.5 Conclusion

Singular Value Decomposition is a powerful mathematical tool with numerous applications in data analysis and machine learning. Understanding how SVD works and how it can be applied is crucial for analyzing and interpreting data efficiently.