SVD Forms Comparison

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About This MicroSim

This infographic compares the three main forms of the Singular Value Decomposition:

1. Full SVD: Complete decomposition with all singular vectors
2. Compact SVD: Only keeps non-zero singular values
3. Truncated SVD: Keeps only the top k singular values

SVD Forms Summary

Form	Matrices	Exact?	Use Case
Full	U(m×m), Σ(m×n), Vᵀ(n×n)	Yes	Complete subspace analysis
Compact	U(m×r), Σ(r×r), Vᵀ(r×n)	Yes	Efficient exact storage
Truncated	U(m×k), Σ(k×k), Vᵀ(k×n)	No	Low-rank approximation

Visual Legend

• Blue blocks: Kept components (contain information)
• Gray blocks: Discarded components (zeros or truncated)
• Light gray: Structural zeros

How to Use

1. Adjust m and n to change matrix dimensions
2. Set rank r to see how many singular values are non-zero
3. Set k to control truncation level
4. Compare storage requirements between forms

Learning Objectives

After using this MicroSim, students will be able to:

• Distinguish between Full, Compact, and Truncated SVD
• Calculate storage requirements for each form
• Choose the appropriate SVD form for a given application
• Understand the tradeoff between accuracy and storage

References

• Chapter 7: Matrix Decompositions - SVD section
• NumPy documentation: np.linalg.svd(full_matrices=False)
• scikit-learn: TruncatedSVD