# svd_sphere

svd_sphere, a MATLAB code which analyzes a linear map of the unit sphere caused by an arbitrary 3x3 matrix A, using the singular value decomposition.

The singular value decomposition has uses in solving overdetermined or underdetermined linear systems, linear least squares problems, data compression, the pseudoinverse matrix, reduced order modeling, and the accurate computation of matrix rank and null space.

The singular value decomposition of an M by N rectangular matrix A has the form

```        A(mxn) = U(mxm) * S(mxn) * V'(nxn)
```
where
• U is an orthogonal matrix, whose columns are the left singular vectors;
• S is a diagonal matrix, whose min(m,n) diagonal entries are the singular values;
• V is an orthogonal matrix, whose columns are the right singular vectors;
Note that the transpose of V is used in the decomposition, and that the diagonal matrix S is typically stored as a vector.

### Languages:

svd_sphere is available in a MATLAB version.

### Related Data and Programs:

svd_basis, a MATLAB code which computes a reduced basis for a collection of data vectors using the SVD.

svd_circle, a MATLAB code which analyzes a linear map of the unit circle caused by an arbitrary 2x2 matrix A, using the singular value decomposition.

svd_faces, a MATLAB code which applies singular value decomposition (SVD) analysis to a set of images.

svd_fingerprint, a MATLAB code which reads a file containing a fingerprint image and uses the Singular Value Decomposition (SVD) to compute and display a series of low rank approximations to the image.

svd_gray, a MATLAB code which reads a gray scale image, computes the Singular Value Decomposition (SVD), and constructs a series of low rank approximations to the image.

svd_snowfall, a MATLAB code which reads a file containing historical snowfall data and analyzes the data with the Singular Value Decomposition (SVD).

svd_test, a MATLAB code which demonstrates the Singular Value Decomposition (SVD) for a simple example.

svd_truncated_test, a MATLAB code which demonstrates the computation of the reduced or truncated Singular Value Decomposition (SVD) that is useful for cases when one dimension of the matrix is much smaller than the other.

### Reference:

1. Lloyd Trefethen, David Bau,
Numerical Linear Algebra,
SIAM, 1997,
ISBN: 0-89871-361-7,
LC: QA184.T74.

### Source Code:

• sphere_sample.m samples points from the unit sphere.
• svd_sphere.m plots the the unit sphere, its image under a map y=A*x, and the images of the left and right singular vectors. y = A*x.

Last revised on 25 January 2019.