Applications

Linear Dynamical Systems

LQR Control

Kalman Filters

Probability

Forbenius Perrod theorem - there is a steady sta probability distribution

Quantum Mechanics

Least Squares Optimization

Boundary Value Problems

Filtering

Fourier transforms Wavelet decompositions PCA

Not sure how to arrange this hierarchy

Matrices

Kronecker products

Numerical

Decompositions

LU

SVD

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

Jordan

QR

Eigenvector

Power method

Krylov

Conjugate Gradient

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Eigenvalues

Resolvent

Characteristic polynomial

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.

Schur Complements

Domain Decomposition

H-Matrices

Infinite domains

Circuit equivalents

Resources

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