[Math]Matrix - Decomposition

. transpose
  . aij와 aji을 바꾸는 연산

. Band matrix
  . 1 < p , q < n
  . aij = 0 if p <= j-i or q <= i-j
  . bandwidth : w = p + q - 1

. strictly diagonally dominant
  . A : n x n
  . |aii| > sigma((j = 1..n, j !=i), |aij|)

. Positive definite
  . It's symmetric
  . And x^T*A*x > 0 for every n-dimensional vector x != 0
  . x^T*A*x = sigma(i=1..n, sigma(j=1..n, aij*xi*xj))
  . Theorem : If A is positive definite matrix
   . A has an inverse
   . aii > 0
   . max 1<=k,j<=n |akj| <= max 1<=i<=n |aii|
   . (aij)^2 < aiiajj, for each i !=j

. leading principal submatrix
  . A : n x n일 때
  . 1<=k<=n 이면 Ak = 가장 처음 k x k 원소만 모은 Matrix

. Diagonal matrix
  . p = 1, q = 1인 band matrix
  . bandwidth = 1

. Tridiagonal matrix
  . p = 2, q = 2인 band matrix
  . bandwidth = 3

. cyclic tridiagonal matrix
  . Tridiagonal matrix에 [1,n], [n,1]에 값을 채워 넣은 것
  . 모든 row가 각각 3개의 원소, 모든 column이 각각 3개의 원소로 구성되어 있다.

. symmetric matrix
  A = A^T

. symmetric cyclic tridiagonal matrix
  ex) closed natural B-spline curve의 control vertex와 

. LU factorization
  . A = LU
  . L : lower triangular matrix
  . U : upper triangular matrix
  . Gaussian Elimination을 이용하면 구할 수 있다.
  . Crout Factorization for tridiagonal linear systems

Matlab에서 LU factorization(decomposition) 하기
X = [[4 1 0 0 1];[1 4 1 0 0];[0 1 4 1 0];[0 0 1 4 1];[1 0 0 1 4]]
[L,U] = lu(X)
inv(U)

maple에서 LU factorization(decomposition) 하기
with(linalg):
A := matrix([[4,1,0,0,1],[1,4,1,0,0],[0,1,4,1,0],[0,0,1,4,1],[1,0,0,1,4]]);
U:=LUdecomp(A);
L:=evalm(A&*inverse(U));
inverse(L);

. QR factorization
  . A = QR
  . R : upper triangular
  . Q^T * Q = I
  . Q^T : Transpose of Q
  . I : identity matrix

. LDL^T Factorization
  . A = LDL^T
  . L : lower triangular matrix with 1's along the diagonal
  . D : diagonal matrix with positive entries on the diagonal.

. Cholesky
  . A = L*L^T
  . L : lower triangular



. 참고서적
  . Linear algebra - 쉬운 것은 개론책 어려운 것은 일반 책을 본다.
  . Calculus, Numerical Analysis 책들에도 기본적인 것들은 언급된다.