You can execute it easily when you are in the [impetuous environment](https://github.com/richardtjornhammar/rixcfgs/blob/master/code/environments/impetuous-shell.nix). Just write
You can execute it easily when you are in the [impetuous environment](https://github.com/richardtjornhammar/rixcfgs/blob/master/code/environments/impetuous-shell.nix). Just write
```
```
...
@@ -495,6 +496,76 @@ The [radial distribution function](https://en.wikipedia.org/wiki/Radial_distribu
...
@@ -495,6 +496,76 @@ The [radial distribution function](https://en.wikipedia.org/wiki/Radial_distribu
The functions `select_from_distance_matrix` uses boolean indexing to select rows and columns (it is symmetric) in the distance matrix and the `exclusive_pdist` function calculates all pairs between the points in the two separate groups.
The functions `select_from_distance_matrix` uses boolean indexing to select rows and columns (it is symmetric) in the distance matrix and the `exclusive_pdist` function calculates all pairs between the points in the two separate groups.
# Example 10: Householder decomposition
In this example we will compare the decompostion of square and rectangular matrices before and after Householder decomposition. We recall that the Householder decomposition is a way of factorising matrices into orthogonal components and a tridiagonal matrix. The routine is implemented in the `impetuous.reducer` module under the name `Householder_reduction`. Now, why is any of that important? The Householder matrices are deterministically determinable and consitutes an unambigous decomposition of your data. The factors are easy to use to further solve what different types of oprations will do you your original matrix. One can, for instance, use it to calculate the ambigous SVD decomposition or calculate eigenvalues for rectangular matrices.
Let us assume that you have a running environment and a set of matrices that you like
```
import numpy as np
import pandas as pd
if __name__=='__main__' :
from impetuous.reducer import ASVD, Householder_reduction
df = lambda x:pd.DataFrame(x)
if True :
B = np.array( [ [ 4 , 1 , -2 , 2 ] ,
[ 1 , 2 , 0 , 1 ] ,
[ -2 , 0 , 3 , -2 ] ,
[ 2 , 1 , -2 , -1 ] ] )
if True :
A = np.array([ [ 22 , 10 , 2 , 3 , 7 ] ,
[ 14 , 7 , 10 , 0 , 8 ] ,
[ -1 , 13 , -1 , -11 , 3 ] ,
[ -3 , -2 , 13 , -2 , 4 ] ,
[ 9 , 8 , 1 , -2 , 4 ] ,
[ 9 , 1 , -7 , 5 , -1 ] ,
[ 2 , -6 , 6 , 5 , 1 ] ,
[ 4 , 5 , 0 , -2 , 2 ] ] )
```
you might notice that the eigenvalues and the singular values of the square matrix `B` look similar
but that the eigenvalues for the Householder reduction of the matrix B and the matrix B are the same
```
HB = Householder_reduction ( B )[1]
print ( np.linalg.eig( B)[0] )
print ( np.linalg.eig(HB)[0] )
```
We readily note that this is also true for the singular values of the matrix `B` and the matrix `HB`. For the rectangular matrix `A` the eigenvalues are not defined when using `numpy`. The `SVD` decomposition is defined and we use it to check if the singular values are the same for the Householder reduction of the matrix A and the matrix A.
```
print ( "BUT THE HOUSEHOLDER REDUCTION IS")
HOUSEH = Householder_reduction ( A )[1]
print ( "SVD ORIGINAL : " , df(np.linalg.svd(A)[1]) )
and lo and behold. They are. So we feel confident that using the eigenvalues from the square part of the Householder matrix (the rest is zero anyway) to calculate the eigenvalues of the rectangular matrix is ok. But wait, why are they complex valued now? :^D
```
n = np.min(np.shape(HOUSEH))
print ( "SVD SKEW H : " , df(np.linalg.svd(HOUSEH)[1]) )
print ( "SVD SQUARE H : " , df(np.linalg.svd(HOUSEH[:n,:n])[1]) )
print ( "SVD ORIGINAL : " , df(np.linalg.svd(A)[1]) )
