# Applications

## Probability

Forbenius Perrod theorem - there is a steady sta probability distribution

## Filtering

Fourier transforms Wavelet decompositions PCA

Not sure how to arrange this hierarchy

# Decompositions

### SVD

https://peterbloem.nl/blog/pca-4

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# Special Matrices

List of named matrices

import numpy as np
import scipy.linalg as linalg
from scipy.linalg import toeplitz
K = toeplitz([2,-1,0,0])
print(K)
print(linalg.inv(K))
from sympy import *
init_printing(use_unicode=True)
Ksym = Matrix(K)
print(Ksym.inv())



# Determinants

Funny funny fellows indeed. Geometrically is the “volume” spanned by the columns. If the matrix represents a transformation, if is the factor of volume shrinkage of the transformation

A definition is an antisymetric recursive one. Why is this formula right? https://en.wikipedia.org/wiki/Laplace_expansion

det([a b; c d]) = ad-bc

Cramer’s rule gives a direct solution to the inverse of a matrix. https://en.wikipedia.org/wiki/Cramer%27s_rule Mainly useful in the 2x2 case

Facts:

1. det(A) = prod of eigvals.
2. det(AB) = det(A)det(B)
3. det(A) = product of pivots in LU form. A more useful way of calulating than brute force

Charactersitic Polynomial = det(A - lam). The roots of this polynomial are eigenvalues.

# Resources

• Horn, Roger A.; Charles R. Johnson (1985). Matrix Analysis.
• COmputational Science and Engineering