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Singular Value Decomposition

  • Understand the concept of Singular Value Decomposition (SVD).
  • Decompose a matrix into orthogonal matrices and singular values.
  • Learn how to compute SVD step by step.
  • Explore applications of SVD in data compression and machine learning.

Definition of Singular Value Decomposition

The **Singular Value Decomposition (SVD)** of an \( m \times n \) matrix \( A \) is a factorization of the form:

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

where:

  • \( U \) is an \( m \times m \) orthogonal matrix (left singular vectors).
  • \( \Sigma \) is an \( m \times n \) diagonal matrix with **singular values** on the diagonal.
  • \( V \) is an \( n \times n \) orthogonal matrix (right singular vectors).

Computing SVD Step by Step

To compute \( A = U \Sigma V^T \):

  1. Find the eigenvalues and eigenvectors of \( A^T A \) to get \( V \).
  2. Find the eigenvalues and eigenvectors of \( A A^T \) to get \( U \).
  3. Compute singular values as \( \sigma_i = \sqrt{\lambda_i} \), where \( \lambda_i \) are the eigenvalues of \( A^T A \).

Proof

Since \( A^T A \) is symmetric, it can be diagonalized:

\[ A^T A = V \Lambda V^T \]

where \( \Lambda \) contains the eigenvalues of \( A^T A \). Taking the square root of \( \Lambda \) gives \( \Sigma \), leading to:

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

Applications of SVD

SVD is widely used in:

  • Data Compression: Reducing dimensionality while preserving essential information.
  • Image Processing: Approximating images with lower-rank matrices.
  • Machine Learning: Reducing feature space in Principal Component Analysis (PCA).
  • Signal Processing: Noise filtering and signal reconstruction.

Examples

Example 1: Find the SVD of:

\[ A = \begin{bmatrix} 4 & 0 \\ 3 & -5 \end{bmatrix} \]

Step 1: Compute \( A^T A \):

\[ A^T A = \begin{bmatrix} 4 & 3 \\ 0 & -5 \end{bmatrix} \begin{bmatrix} 4 & 0 \\ 3 & -5 \end{bmatrix} = \begin{bmatrix} 25 & -15 \\ -15 & 25 \end{bmatrix} \]

Step 2: Compute eigenvalues of \( A^T A \):

\[ \det \begin{bmatrix} 25 - \lambda & -15 \\ -15 & 25 - \lambda \end{bmatrix} = 0 \] \[ (25 - \lambda)^2 - 225 = 0 \] \[ \lambda^2 - 50\lambda + 400 = 0 \] \[ \lambda = 40, 10 \]

Step 3: Compute singular values:

\[ \sigma_1 = \sqrt{40}, \quad \sigma_2 = \sqrt{10} \]

Step 4: Compute eigenvectors for \( V \), then \( U \), and form the decomposition.

Exercises

  • Question 1: Compute the singular values for \( A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \).
  • Question 2: Find the SVD of \( A = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix} \).
  • Question 3: Explain why \( U \) and \( V \) are always orthogonal in SVD.

  • Answer 1: \( \sigma_1 = \sqrt{5}, \sigma_2 = \sqrt{5} \).
  • Answer 2: \( U = I \), \( \Sigma = \begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix} \), \( V = I \).
  • Answer 3: Because the eigenvectors of \( A^T A \) and \( A A^T \) form orthonormal bases.

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