Lecture Notes: Matrix Rank, Image Compression, and SVD

Introduction

• The video is sponsored by Skillshare.

Rank One Matrix

• Definition: A rank one matrix is a matrix where rows/columns are constant multiples of the first row/column.
  • Example: If rows and columns are multiples of each other (e.g., second row twice the first).
• Matrix Form: Can be expressed as the product of a row matrix and a column matrix using matrix multiplication.
• Data Compression: A 3x3 rank one matrix (9 numbers) can be stored as a 3+3 (6 numbers).
  • 10x10 rank one matrix: 100 numbers to 20 numbers (10+10).

Sum of Rank One Matrices

• Key Concept: Any matrix can be decomposed into a sum of rank one matrices.
• Singular Value Decomposition (SVD): Fundamental in linear algebra for matrix decomposition.

Image Compression

• Digital Images as Matrices: Images are matrices with pixel values as entries.
  • Example: 1000x1000 pixel image corresponds to a 1000x1000 matrix.
• Compression Technique: Represent the image matrix as a sum of rank one matrices and discard some.
  • Example: A 5x5 image of number 4 can be compressed using 3 rank one matrices.
  • Discarding matrices reduces the number of stored values (e.g., 25 numbers to 20 numbers).
  • Further compression (e.g., using only 1 rank one matrix) can result in data loss.

Singular Value Decomposition (SVD)

• Matrix Factorization: Matrix A can be factored into matrices U, Σ (Sigma), and Vᵀ (V transpose).
• Sigma: Diagonal or pseudo-diagonal matrix indicating weight of rank one matrices.
• Expression: Matrix A = UΣVᵀ can be re-expressed as a sum of weighted rank one matrices using Σ.
  • Each term in the sum corresponds to a singular value in Σ multiplied by respective rows and columns of U and Vᵀ.

Application in Image Compression

• Weighting Rank One Matrices: Rank one matrices are weighted by singular values in Σ.
• Compression Strategy: Discard matrices with smallest weights for compression.
  • Example: In image compression, smallest weighted matrices are removed (e.g., those with smallest Σ values).

Skillshare Promotion

• Skillshare: An online learning platform with classes on various topics including programming, data science, etc.
• Offer: First 500 users via provided link receive a one-month free trial.

Conclusion

• Image compression leverages mathematical concepts like rank one matrices and SVD to reduce data storage while maintaining essential details.