What is the Geometric Interpretation of the Final Step in SVD?

Click For Summary
SUMMARY

The discussion centers on the geometric interpretation of the final step in Singular Value Decomposition (SVD), specifically how the matrix U transforms the output from the stretching operation performed by Σ back into R^m. The initial steps involve V^T rotating the input vector x in R^n, followed by Σ stretching the result. The confusion arises in understanding that while U does not serve as the inverse of V^T, it effectively maps the stretched vector into the original space of R^m, completing the transformation process.

PREREQUISITES
  • Understanding of Singular Value Decomposition (SVD)
  • Familiarity with matrix transformations in linear algebra
  • Knowledge of eigenvalue decomposition
  • Basic concepts of vector spaces in R^n and R^m
NEXT STEPS
  • Study the geometric interpretations of Singular Value Decomposition (SVD)
  • Learn about matrix transformations and their effects on vector spaces
  • Explore the relationship between eigenvalue decomposition and SVD
  • Investigate applications of SVD in data compression and dimensionality reduction
USEFUL FOR

Mathematicians, data scientists, and anyone studying linear algebra or applying SVD in machine learning and data analysis.

daviddoria
Messages
96
Reaction score
0
<br /> Ax = U \Sigma V^T x<br />

(A is an m by n matrix)

I understand the first two steps,

1) V^T takes x and expresses it in a new basis in R^n (since x is already in R^n, this is simply a rotation)

2) \Sigma takes the result of (1) and stretches it

The third step is where I'm a bit fuzzy...

3) U takes the result of (2) and puts it into R^m. In eigenvalue decomposition, this is just the inverse transformation of V^T, but I always read "this is not the inverse transformation of step 1"

Can someone clarify this last step a bit for me?

Thanks!

Dave
 
Physics news on Phys.org

Similar threads

Undergrad Which Are the EOFs in SVD?
  • · Replies 1 ·
Replies
1
Views
1K
Graduate The Power of SVD in Data Processing: Insights and Examples
  • · Replies 10 ·
Replies
10
Views
4K
Undergrad Interpretation of "pseudo-diagonalisation"
  • · Replies 5 ·
Replies
5
Views
2K
MHB Geometric interpretation of maps
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Grover algorithm geometric interpretation
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Passive Transformation and Rotation Matrix
  • · Replies 1 ·
Replies
1
Views
4K
Graduate Geometric insight about rotations needed
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Proving Convexity of the Set X = {(x, y) E R^2; ax + by <= c} in R^2
  • · Replies 5 ·
Replies
5
Views
2K
Graduate What Happens to Av^j When j Exceeds Rank r in SVD?
  • · Replies 4 ·
Replies
4
Views
3K