. 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.