# svd_sphere

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

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 and a Python version.

### Related Data and Programs:

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

svd_lls, a Python code which uses the singular value decomposition (SVD) to construct and plot the best affine and linear relationships in the sense of least square, between two vectors of data.

svd_powers, a Python code which applies singular value decomposition (SVD) analysis to powers x(i)^(j-1).

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

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

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

svd_truncated_test, a Python 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:

Last revised on 03 April 2022.