MATLAB求解線性方程的過程基于三種分解法則：
（１）Cholesky分解，針對(duì)對(duì)稱正定矩陣；
（２）高斯消元法， 針對(duì)一般矩陣；
（３）正交化， 針對(duì)一般矩陣（行數(shù)≠列數(shù)）
這三種分解運(yùn)算分別由chol, lu和 qr三個(gè)函數(shù)來分解.
1． Cholesky分解(Cholesky Decomposition)
僅適用于對(duì)稱和上三角矩陣
例：cholesky分解。
a=pascal(6)
b=chol(a)
a =
1 1 1 1 1 1
1 2 3 4 5 6
1 3 6 10 15 21
1 4 10 20 35 56
1 5 15 35 70 126
1 6 21 56 126 252
b =
1 1 1 1 1 1
0 1 2 3 4 5
0 0 1 3 6 10
0 0 0 1 4 10
0 0 0 0 1 5
0 0 0 0 0 1
CHOL Cholesky factorization.
CHOL(X) uses only the diagonal and upper triangle of X. The lower triangular is assumed to be the (complex conjugate) transpose of the upper. If X is positive definite, then R = CHOL(X) produces an upper triangular R so that R'*R = X. If X is not positive definite, an error message is printed.
[R,p] = CHOL(X), with two output arguments, never produces an
error message. If X is positive definite, then p is 0 and R is the same as above. But if X is not positive definite, then p is a positive integer.
When X is full, R is an upper triangular matrix of order q = p-1
so that R'*R = X(1:q,1:q). When X is sparse, R is an upper triangular matrix of size q-by-n so that the L-shaped region of the first q rows and first q columns of R'*R agree with those of X.
2． LU分解(LU factorization).
用lu函數(shù)完成LU分解，將矩陣分解為上、下兩個(gè)三角陣，其調(diào)用格式為：
[l,u]=lu(a) l代表下三角陣，u代表上三角陣。
例：
LU分解。
a=[47 24 22; 11 44 0;30 38 41]
[l,u]=lu(a)
a =
47 24 22
11 44 0
30 38 41
l =
1.0000 0 0
0.2340 1.0000 0
0.6383 0.5909 1.0000
u =
47.0000 24.0000 22.0000
0 38.3830 -5.1489
0 0 30.0000
LU LU factorization.
[L,U] = LU(X) stores an upper triangular matrix in U and a "psychologically lower triangular matrix" (i.e. a product of lower triangular and permutation matrices) in L, so that X = L*U. X can be rectangular.
[L,U,P] = LU(X) returns unit lower triangular matrix L, upper triangular matrix U, and permutation matrix P so that P*X = L*U.
3． QR分解(Orthogonal-triangular decomposition).
函數(shù)調(diào)用格式：[q,r]=qr(a), q代表正規(guī)正交矩陣，r代表三角形矩陣。原始陣a不必一定是方陣。如果矩陣a是m×n階的，則矩陣q是m×m階的，矩陣r是m×n階的。
例：QR分解.
A=[22 46 20 20; 30 36 46 44;39 8 45 2];
[q,r]=qr(A)
q =
-0.4082 -0.7209 -0.5601
-0.5566 -0.2898 0.7786
-0.7236 0.6296 -0.2829
r =
-53.8981 -44.6027 -66.3289 -34.1014
0 -38.5564 0.5823 -25.9097
0 0 11.8800 22.4896
QR Orthogonal-triangular decomposition.
[Q,R] = QR(A) produces an upper triangular matrix R of the same
dimension as A and a unitary matrix Q so that A = Q*R.
[Q,R,E] = QR(A) produces a permutation matrix E, an upper
triangular R and a unitary Q so that A*E = Q*R. The column
permutation E is chosen so that abs(diag(R)) is decreasing.
[Q,R] = QR(A,0) produces the "economy size" decomposition. If A is m-by-n with m > n, then only the first n columns of Q are computed.
4. 特征值與特征矢量(Eigenvalues and eigenvectors).
MATLAB中使用函數(shù)eig計(jì)算特征值和 特征矢量，有兩種調(diào)用方法：
*e=eig(a), 其中e是包含特征值的矢量；
*[v,d]=eig(a), 其中v是一個(gè)與a相同的n×n階矩陣，它的每一列是矩陣a的一個(gè)特征值所對(duì)應(yīng)的特征矢量，d為對(duì)角陣，其對(duì)角元素即為矩陣a的特征值。
例：計(jì)算特征值和特征矢量。
a=[34 25 15; 18 35 9; 41 21 9]
e=eig(a)
[v,d]=eig(a)
a =
34 25 15
18 35 9
41 21 9
e =
68.5066
15.5122
-6.0187
v =
-0.6227 -0.4409 -0.3105
-0.4969 0.6786 -0.0717
-0.6044 -0.5875 0.9479
d =
68.5066 0 0
0 15.5122 0
0 0 -6.0187
EIG Eigenvalues and eigenvectors.
E = EIG(X) is a vector containing the eigenvalues of a square matrix X.
[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D.
[V,D] = EIG(X,'nobalance') performs the computation with balancing
disabled, which sometimes gives more accurate results for certain
problems with unusual scaling. If X is symmetric, EIG(X,'nobalance')
is ignored since X is already balanced.
5. 奇異值分解.( Singular value decomposition).
如存在兩個(gè)矢量u,v及一常數(shù)c,使得矩陣A滿足：Av=cu, A’u=cv
稱c為奇異值，稱u,v為奇異矢量。
將奇異值寫成對(duì)角方陣∑，而相對(duì)應(yīng)的奇異矢量作為列矢量則可寫成兩個(gè)正交矩陣U，V， 使得： AV=U∑， A‘U=V∑ 因?yàn)閁，V正交，所以可得奇異值表達(dá)式：
A=U∑V’。
一個(gè)m行n列的矩陣A經(jīng)奇異值分解，可求得m行m列的U, m行n列的矩陣∑和n行n列的矩陣V.。
奇異值分解用svd函數(shù)實(shí)現(xiàn)，調(diào)用格式為；
[u,s,v]=svd(a)
SVD Singular value decomposition.
[U,S,V] = SVD(X) produces a diagonal matrix S, of the same dimension as X and with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that X = U*S*V'.
S = SVD(X) returns a vector containing the singular values.
[U,S,V] = SVD(X,0) produces the "economy size" decomposition. If X is m-by-n with m > n, then only the first n columns of U are computed and S is n-by-n.
例： 奇異值分解。
a=[8 5; 7 3;4 6];
[u,s,v]=svd(a) % s為奇異值對(duì)角方陣
u =
-0.6841 -0.1826 -0.7061
-0.5407 -0.5228 0.6591
-0.4895 0.8327 0.2589
s =
13.7649 0
0 3.0865
0 0
v =
-0.8148 -0.5797
-0.5797 0.8148
