Geometric interpretation of SVD

In summary, the singular value decomposition (SVD) is a decomposition of a matrix A into three matrices: U, Σ, and V^T. The first two steps involve transforming the original matrix x into a new basis in R^n and then stretching it. The third step, performed by matrix U, puts the result into R^m. This step is not the inverse transformation of step 1, but it has a similar effect in eigenvalue decomposition. The Wikipedia entry for SVD provides more detailed information.
  • #1
daviddoria
97
0
[tex]
Ax = U \Sigma V^T x
[/tex]

(A is an m by n matrix)

I understand the first two steps,

1) [tex]V^T[/tex] 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) [tex]\Sigma[/tex] 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

1. What is the purpose of SVD in geometric interpretation?

SVD, or Singular Value Decomposition, is a mathematical tool used to decompose a matrix into three components: a left singular matrix, a diagonal matrix of singular values, and a right singular matrix. In geometric interpretation, SVD helps us understand the structure and properties of a matrix in terms of rotations and scalings along orthogonal axes.

2. How does SVD relate to principal component analysis (PCA)?

PCA is a statistical technique used to reduce the dimensionality of data while preserving the most important information. SVD is a key component of PCA, as it allows us to decompose the data matrix into its principal components, which are the orthogonal axes that capture the most variance in the data.

3. Can SVD be applied to non-square matrices?

Yes, SVD can be applied to non-square matrices. In fact, it is commonly used for rectangular matrices in data analysis tasks such as image compression and recommender systems. The resulting decomposition will have a different form, with the left and right singular matrices being rectangular instead of square.

4. How does the number of singular values affect the geometric interpretation of SVD?

The number of singular values determines the number of principal components or dimensions in the matrix. The larger the number of singular values, the more information is preserved and the more accurate the geometric interpretation will be. However, having too many singular values can also lead to overfitting and loss of generalizability.

5. Can SVD be used for data compression?

Yes, SVD is commonly used for data compression by reducing the dimensionality of the data while preserving most of its important information. By keeping only the most significant singular values, we can reduce the size of the matrix and still retain a good approximation of the original data.

Similar threads

I Dimension and solution to matrix-vector product
    • Linear and Abstract Algebra
Replies
4
Views
879
I How to Compute the EOFs by SVD?
    • Linear and Abstract Algebra
Replies
1
Views
1K
Vectors as geometric objects and vectors as any mathematical objects
    • Linear and Abstract Algebra
Replies
27
Views
1K
I Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
    • Linear and Abstract Algebra
Replies
4
Views
1K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
    • Linear and Abstract Algebra
Replies
3
Views
2K
Unlocking the Power of SVD in Data Processing: Insights and Examples
    • Linear and Abstract Algebra
Replies
10
Views
3K
I Proving Convexity of the Set X = {(x, y) E R^2; ax + by <= c} in R^2
    • Linear and Abstract Algebra
Replies
5
Views
1K
I Interpretation of "pseudo-diagonalisation"
    • Linear and Abstract Algebra
Replies
5
Views
1K
SVD of a reduced rank matrix still has non-zero U and V`?
    • Linear and Abstract Algebra
Replies
2
Views
1K
MHB Diagonalizable transformation - Existence of basis
    • Linear and Abstract Algebra
2
Replies
52
Views
2K

Hot Threads

Recent Insights

Back
Top