📚 Topic Summary
Singular Value Decomposition (SVD) is a powerful technique in linear algebra that decomposes any matrix into three simpler matrices: $U$, $\Sigma$, and $V^T$. SVD reveals essential structural information about the original matrix, such as its rank and the principal components of the data it represents. It's like breaking down a complex puzzle into smaller, more manageable pieces.
SVD has applications in data compression, noise reduction, image processing, and recommender systems. By focusing on the largest singular values in $\Sigma$, we can approximate the original matrix while reducing storage requirements and computational complexity. Understanding both the theory and the practical implementation of SVD unlocks a wide range of problem-solving capabilities.
🧠 Part A: Vocabulary
Match the term to its correct definition:
| Term |
Definition |
| 1. Singular Value |
A. A matrix whose columns are orthonormal eigenvectors of $A^TA$. |
| 2. Orthonormal |
B. A measure of the 'energy' captured by a principal component. |
| 3. Principal Component |
C. Vectors that are orthogonal (perpendicular) and have a length of 1. |
| 4. $U$ matrix |
D. A matrix whose columns are orthonormal eigenvectors of $AA^T$. |
| 5. $V$ matrix |
E. The direction in data that explains the most variance. |
(Answers: 1-B, 2-C, 3-E, 4-D, 5-A)
✏️ Part B: Fill in the Blanks
SVD decomposes a matrix $A$ into three matrices: $U$, $\Sigma$, and ____. The matrix $\Sigma$ is a ____ matrix containing ____ values on its diagonal. The singular values are always ____ or equal to zero.
(Answers: $V^T$, diagonal, singular, greater than)
💡 Part C: Critical Thinking
Explain how SVD can be used to reduce noise in an image. Provide a concrete example.