21.16. Singular Value Decomposition (SVD)

Try these first

1. 1.

   What does SVD stand for?

   Hint

   Think about the components it factors into.

2. 2.

   What is the type of matrix that contains the singular values in SVD?

   Hint

   This matrix has non-zero entries only along the diagonal.

3. 3.

   What is the formula for Singular Value Decomposition?

   • A = ABC
   • A = UΣV^T
   • A = UVΣ

   Hint

   Recall the notation of SVD.

4. 4.

   Is the matrix Σ in SVD a diagonal matrix?

   • True
   • False

   Hint

   What kind of entries does a diagonal matrix have?

5. 5.

   Consider a matrix A. Derive its SVD step-by-step, explaining each step's significance.

   Hint

   Visualize each step as layers building up the understanding of the matrix's structure.

6. 6.

   Provide a real-world dataset and perform PCA using SVD to reduce dimensions. Explain the retained singular values' significance.

   Hint

   Think about how the information is expressed dimensionally in the context of real-world applications.