主要包括以下内容:
 创建矩阵向量;矩阵加减,乘积;矩阵的逆;行列式的值;特征值与特征向量;QR分解;奇异值分解;广义逆;backsolve与fowardsolve函数;取矩阵的上下三角元素;向量化算子等. 
1  
 在R中可以用函数c()来创建一个向量,例如:
> x=c(1,2,3,4)
 > x
 [1] 1 2 3 4
2  
 在R中可以用函数matrix()来创建一个矩阵,应用该函数时需要输入必要的参数值。
> args(matrix)
 function (data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL)
data项为必要的矩阵元素,nrow为行数,ncol为列数,注意nrow与ncol的乘积应为矩阵元素个数,byrow项控制排列元素时是否按行进行,dimnames给定行和列的名称。例如:
> matrix(1:12,nrow=3,ncol=4)
     [,1] [,2] [,3] [,4]
 [1,]   1   4   7   10
 [2,]   2   5   8   11
 [3,]   3   6   9   12
 > matrix(1:12,nrow=4,ncol=3)
     [,1] [,2] [,3]
 [1,]   1   5   9
 [2,]   2   6   10
 [3,]   3   7   11
 [4,]   4   8   12
 > matrix(1:12,nrow=4,ncol=3,byrow=T)
     [,1] [,2] [,3]
 [1,]   1   2   3
 [2,]   4   5   6
 [3,]   7   8   9
 [4,]   10   11   12
> rowname
 [1] "r1" "r2" "r3"
 > colname=c("c1","c2","c3","c4")
 > colname
 [1] "c1" "c2" "c3" "c4"
 > matrix(1:12,nrow=3,ncol=4,dimnames=list(rowname,colname))
   c1 c2 c3 c4
 r1 1 4 7 10
 r2 2 5 8 11
3  
 A为m×n矩阵,求A'在R中可用函数t(),例如:
 >A=matrix(1:12,nrow=3,ncol=4)
 > A
    [,1] [,2] [,3] [,4]
 [1,]   1   4   7   10
 [2,]   2   5   8   11
 [3,]   3   6   9   12
 > t(A)
    [,1] [,2] [,3]
 [1,]   1   2   3
 [2,]   4   5   6
 [3,]   7   8   9
 [4,]   10   11   12
若将函数t()作用于一个向量x,则R默认x为列向量,返回结果为一个行向量,例如:
 >x
 [1] 1 2 3 4 5 6 7 8 9 10
 > t(x)
   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
 [1,]   1   2   3   4   5   6   7   8   9   10
 > class(x)
 [1] "integer"
 > class(t(x))
 [1] "matrix"
 若想得到一个列向量,可用t(t(x)),例如:
 >x
 [1] 1 2 3 4 5 6 7 8 9 10
 > t(t(x))
     [,1]
 [1,]   1
 [2,]   2
 [3,]   3
 [4,]   4
 [5,]   5
 [6,]   6
 [7,]   7
 [8,]   8
 [9,]   9
 [10,]  10
 > y=t(t(x))
 > t(t(y))
     [,1]
 [1,]   1
 [2,]   2
 [3,]   3
 [4,]   4
 [5,]   5
 [6,]   6
 [7,]   7
 [8,]   8
 [9,]   9
 [10,]   10
4  矩阵相加减
 在R中对同行同列矩阵相加减,可用符号:“+”、“-”,例如:
> A=B=matrix(1:12,nrow=3,ncol=4)
 > A+B
     [,1] [,2] [,3] [,4]
 [1,]   2   8   14   20
 [2,]   4   10   16   22
 [3,]   6   12   18   24
 > A-B
    [,1] [,2] [,3] [,4]
 [1,]   0   0   0   0
 [2,]   0   0   0   0
 [3,]   0   0   0   0
5 数与矩阵相乘
 A为m×n矩阵,c>0,在R中求cA可用符号:“*”,例如:
> c=2
 > c*A
     [,1] [,2] [,3] [,4]
 [1,]   2   8   14   20
 [2,]   4   10  16   22
 [3,]   6   12  18  
6  
 A为m×n矩阵,B为n×k矩阵,在R中求AB可用符号:“%*%”,例如:
> A=matrix(1:12,nrow=3,ncol=4)
 > B=matrix(1:12,nrow=4,ncol=3)
 > A%*%B
     [,1] [,2] [,3]
 [1,]   70  158 246
 [2,]   80  184 288
 [3,]   90 
 若A为n×m矩阵,要得到A'B,可用函数crossprod(),该函数计算结果与t(A)%*%B相同,但是效率更高。例如:
> A=matrix(1:12,nrow=4,ncol=3)
 > B=matrix(1:12,nrow=4,ncol=3)
 > t(A)%*%B
     [,1] [,2] [,3]
 [1,]  30   70 110
 [2,]  70  174 278
 [3,] 110  278 446
 > crossprod(A,B)
     [,1] [,2] [,3]
 [1,]  30  70 110
 [2,]  70 174 278
 [3,] 110 278 446
矩阵Hadamard积:若A={aij}m×n, B={bij}m×n, 则矩阵的Hadamard积定义为:
 A⊙B={aij bij }m×n,R中Hadamard积可以直接运用运算符“*”例如:
> A=matrix(1:16,4,4)
 > A
     [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12   16
 > B=A
 > A*B
     [,1] [,2] [,3] [,4]
 [1,]   1   25   81 169
 [2,]   4   36 100 196
 [3,]   9   49 121 225
 [4,]   16   64 144 256
R中这两个运算符的区别区加以注意。
7  
 例如要取一个方阵的对角元素,
> A=matrix(1:16,nrow=4,ncol=4)
 > A
     [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12   16
 > diag(A)
 [1] 1 6 11 16
对一个向量应用diag()函数将产生以这个向量为对角元素的对角矩阵,例如:
> diag(diag(A))
     [,1] [,2] [,3] [,4]
 [1,]   1   0   0   0
 [2,]   0   6   0   0
 [3,]   0   0   11   0
 [4,]   0   0   0  
 对一个正整数z应用diag()函数将产生以z维单位矩阵,例如:
> diag(3)
     [,1] [,2] [,3]
 [1,]   1   0   0
 [2,]   0   1   0
 [3,]   0   0  
8  
 矩阵求逆可用函数solve(),应用solve(a, b)运算结果是解线性方程组ax = b,若b缺省,则系统默认为单位矩阵,因此可用其进行矩阵求逆,例如:
> a=matrix(rnorm(16),4,4)
 > a
             [,1]     [,2]     [,3]     [,4]
 [1,] 1.6986019   0.5239738 0.2332094 0.3174184
 [2,] -0.2010667 1.0913013 -1.2093734   0.8096514
 [3,] -0.1797628 -0.7573283 0.2864535 1.3679963
 [4,] -0.2217916 -0.3754700 0.1696771 -1.2424030
 > solve(a)
               [,1]     [,2]     [,3]     [,4]
 [1,] 0.9096360 0.54057479 0.7234861 1.3813059
 [2,] -0.6464172 -0.91849017 -1.7546836 -2.6957775
 [3,] -0.7841661 -1.78780083 -1.5795262 -3.1046207
 [4,] -0.0741260 -0.06308603 0.1854137 -0.6607851
 > solve (a) %*%a
                 [,1]       [,2]           [,3]       [,4]
 [1,] 1.000000e+00 2.748453e-17 -2.787755e-17 -8.023096e-17
 [2,] 1.626303e-19 1.000000e+00 -4.960225e-18 6.977925e-16
 [3,] 2.135878e-17 -4.629543e-17 1.000000e+00 6.201636e-17
 [4,] 1.866183e-17 1.563962e-17 1.183813e-17 1.000000e+00
9  矩阵的特征值与特征向量
 矩阵A的谱分解为A=UΛU',其中Λ是由A的特征值组成的对角矩阵,U的列为A的特征值对应的特征向量,在R中可以用函数eigen()函数得到U和Λ,
> args(eigen)
 function (x, symmetric, only.values = FALSE, EISPACK = FALSE)
其中:x为矩阵,symmetric项指定矩阵x是否为对称矩阵,若不指定,系统将自动检测x是否为对称矩阵。例如:
 > A=diag(4)+1
 > A
   [,1] [,2] [,3] [,4]
 [1,]   2   1   1   1
 [2,]   1   2   1   1
 [3,]   1   1   2   1
 [4,]   1   1   1   2
 > A.eigen=eigen(A,symmetric=T)
 > A.eigen
 $values
 [1] 5 1 1 1

 $vectors
         [,1]     [,2]       [,3]     [,4]
 [1,] 0.5 0.8660254 0.000000e+00 0.0000000
 [2,] 0.5 -0.2886751 -6.408849e-17 0.8164966
 [3,] 0.5 -0.2886751 -7.071068e-01 -0.4082483
 [4,] 0.5 -0.2886751 7.071068e-01 -0.4082483

 > A.eigen$vectors%*%diag(A.eigen$values)%*%t(A.eigen$vectors)
   [,1] [,2] [,3] [,4]
 [1,]   2   1   1   1
 [2,]   1   2   1   1
 [3,]   1   1   2   1
 [4,]   1   1   1   2
 > t(A.eigen$vectors)%*%A.eigen$vectors
             [,1]       [,2]         [,3]         [,4]
 [1,] 1.000000e+00 4.377466e-17 1.626303e-17 -5.095750e-18
 [2,] 4.377466e-17 1.000000e+00 -1.694066e-18 6.349359e-18
 [3,] 1.626303e-17 -1.694066e-18 1.000000e+00 -1.088268e-16
 [4,] -5.095750e-18 6.349359e-18 -1.088268e-16 1.000000e+00
10  
   对于正定矩阵A,可对其进行Choleskey分解,即:A=P'P,其中P为上三角矩阵,在R中可以用函数chol()进行Choleskey分解,例如:
> A
   [,1] [,2] [,3] [,4]
 [1,]   2   1   1   1
 [2,]   1   2   1   1
 [3,]   1   1   2   1
 [4,]   1   1   1   2
 > chol(A)
         [,1]     [,2]     [,3]     [,4]
 [1,] 1.414214 0.7071068 0.7071068 0.7071068
 [2,] 0.000000 1.2247449 0.4082483 0.4082483
 [3,] 0.000000 0.0000000 1.1547005 0.2886751
 [4,] 0.000000 0.0000000 0.0000000 1.1180340
 > t(chol(A))%*%chol(A)
   [,1] [,2] [,3] [,4]
 [1,]   2   1   1   1
 [2,]   1   2   1   1
 [3,]   1   1   2   1
 [4,]   1   1   1   2
 > crossprod(chol(A),chol(A))
   [,1] [,2] [,3] [,4]
 [1,]   2   1   1   1
 [2,]   1   2   1   1
 [3,]   1   1   2   1
 [4,]   1   1   1   2
若矩阵为对称正定矩阵,可以利用Choleskey分解求行列式的值,如:
> prod(diag(chol(A))^2)
 [1] 5
 > det(A)
 [1] 5
若矩阵为对称正定矩阵,可以利用Choleskey分解求矩阵的逆,这时用函数chol2inv(),这种用法更有效。如:
> chol2inv(chol(A))
       [,1] [,2] [,3] [,4]
 [1,] 0.8 -0.2 -0.2 -0.2
 [2,] -0.2 0.8 -0.2 -0.2
 [3,] -0.2 -0.2 0.8 -0.2
 [4,] -0.2 -0.2 -0.2 0.8
 > solve(A)
   [,1] [,2] [,3] [,4]
 [1,] 0.8 -0.2 -0.2 -0.2
 [2,] -0.2 0.8 -0.2 -0.2
 [3,] -0.2 -0.2 0.8 -0.2
 [4,] -0.2 -0.2 -0.2 0.8
11  
   A为m×n矩阵,rank(A)= r, 可以分解为:A=UDV',其中U'U=V'V=I。在R中可以用函数scd()进行奇异值分解,例如:
> A=matrix(1:18,3,6)
 > A
   [,1] [,2] [,3] [,4] [,5] [,6]
 [1,]   1   4   7   10   13   16
 [2,]   2   5   8   11   14   17
 [3,]   3   6   9   12   15   18
 > svd(A)
 $d
 [1] 4.589453e+01 1.640705e+00 3.627301e-16
   $u
           [,1]     [,2]     [,3]
 [1,] -0.5290354 0.74394551 0.4082483
 [2,] -0.5760715 0.03840487 -0.8164966
 [3,] -0.6231077 -0.66713577 0.4082483
 $v
           [,1]     [,2]     [,3]
 [1,] -0.07736219 -0.7196003 -0.18918124
 [2,] -0.19033085 -0.5089325 0.42405898
 [3,] -0.30329950 -0.2982646 -0.45330031
 [4,] -0.41626816 -0.0875968 -0.01637004
 [5,] -0.52923682 0.1230711 0.64231130
 [6,] -0.64220548 0.3337389 -0.40751869
 > A.svd=svd(A)
 > A.svd$u%*%diag(A.svd$d)%*%t(A.svd$v)
   [,1] [,2] [,3] [,4] [,5] [,6]
 [1,]   1   4   7   10   13   16
 [2,]   2   5   8   11   14   17
 [3,]   3   6   9   12   15   18
 > t(A.svd$u)%*%A.svd$u
             [,1]       [,2]       [,3]
 [1,] 1.000000e+00 -1.169312e-16 -3.016793e-17
 [2,] -1.169312e-16 1.000000e+00 -3.678156e-17
 [3,] -3.016793e-17 -3.678156e-17 1.000000e+00
 > t(A.svd$v)%*%A.svd$v
         [,1]       [,2]       [,3]
 [1,] 1.000000e+00 8.248068e-17 -3.903128e-18
 [2,] 8.248068e-17 1.000000e+00 -2.103352e-17
 [3,] -3.903128e-18 -2.103352e-17 1.000000e+00
12  矩阵QR分解
 A为m×n矩阵可以进行QR分解,A=QR,其中:Q'Q=I,在R中可以用函数qr()进行QR分解,例如:
> A=matrix(1:16,4,4)
 > qr(A)
 $qr
       [,1]     [,2]       [,3]       [,4]
 [1,] -5.4772256 -12.7801930 -2.008316e+01 -2.738613e+01
 [2,] 0.3651484 -3.2659863 -6.531973e+00 -9.797959e+00
 [3,] 0.5477226 -0.3781696 2.641083e-15 2.056562e-15
 [4,] 0.7302967 -0.9124744 8.583032e-01 -2.111449e-16

 $rank
 [1] 2

 $qraux
 [1] 1.182574e+00 1.156135e+00 1.513143e+00 2.111449e-16

 $pivot
 [1] 1 2 3 4

 attr(,"class")
 [1] "qr"
rank项返回矩阵的秩,qr项包含了矩阵Q和R的信息,要得到矩阵Q和R,可以用函数qr.Q()和qr.R()作用qr()的返回结果,例如:
> qr.R(qr(A))
       [,1]     [,2]       [,3]       [,4]
 [1,] -5.477226 -12.780193 -2.008316e+01 -2.738613e+01
 [2,] 0.000000 -3.265986 -6.531973e+00 -9.797959e+00
 [3,] 0.000000   0.000000 2.641083e-15 2.056562e-15
 [4,] 0.000000   0.000000 0.000000e+00 -2.111449e-16
 > qr.Q(qr(A))
       [,1]       [,2]     [,3]     [,4]
 [1,] -0.1825742 -8.164966e-01 -0.4000874 -0.37407225
 [2,] -0.3651484 -4.082483e-01 0.2546329 0.79697056
 [3,] -0.5477226 -8.131516e-19 0.6909965 -0.47172438
 [4,] -0.7302967 4.082483e-01 -0.5455419 0.04882607
 > qr.Q(qr(A))%*%qr.R(qr(A))
   [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12   16
 > t(qr.Q(qr(A)))%*%qr.Q(qr(A))
         [,1]       [,2]       [,3]       [,4]
 [1,] 1.000000e+00 -1.457168e-16 -6.760001e-17 -7.659550e-17
 [2,] -1.457168e-16 1.000000e+00 -4.269046e-17 7.011739e-17
 [3,] -6.760001e-17 -4.269046e-17 1.000000e+00 -1.596437e-16
 [4,] -7.659550e-17 7.011739e-17 -1.596437e-16 1.000000e+00
 > qr.X(qr(A))
   [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12   16
13  
   n×m矩阵A+称为m×n矩阵A的Moore-Penrose逆,如果它满足下列条件:
 ①   A A+A=A;②A+A A+= A+;③(A A+)H=A A+;④(A+A)H= A+A
 在R的MASS包中的函数ginv()可计算矩阵A的Moore-Penrose逆,例如:
library(“MASS”)
 > A
   [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12   16
 > ginv(A)
     [,1]   [,2] [,3]   [,4]
 [1,] -0.285 -0.1075 0.07 0.2475
 [2,] -0.145 -0.0525 0.04 0.1325
 [3,] -0.005 0.0025 0.01 0.0175
 [4,] 0.135 0.0575 -0.02 -0.0975
 验证性质1:
> A%*%ginv(A)%*%A
   [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12  
 验证性质2:
> ginv(A)%*%A%*%ginv(A)
     [,1]   [,2] [,3]   [,4]
 [1,] -0.285 -0.1075 0.07 0.2475
 [2,] -0.145 -0.0525 0.04 0.1325
 [3,] -0.005 0.0025 0.01 0.0175
 [4,] 0.135 0.0575 -0.02 -0.0975
 验证性质3:
> t(A%*%ginv(A))
   [,1] [,2] [,3] [,4]
 [1,] 0.7 0.4 0.1 -0.2
 [2,] 0.4 0.3 0.2 0.1
 [3,] 0.1 0.2 0.3 0.4
 [4,] -0.2 0.1 0.4 0.7
 > A%*%ginv(A)
   [,1] [,2] [,3] [,4]
 [1,] 0.7 0.4 0.1 -0.2
 [2,] 0.4 0.3 0.2 0.1
 [3,] 0.1 0.2 0.3 0.4
 [4,] -0.2 0.1 0.4 0.7
 验证性质4:
> t(ginv(A)%*%A)
   [,1] [,2] [,3] [,4]
 [1,] 0.7 0.4 0.1 -0.2
 [2,] 0.4 0.3 0.2 0.1
 [3,] 0.1 0.2 0.3 0.4
 [4,] -0.2 0.1 0.4 0.7
 > ginv(A)%*%A
   [,1] [,2] [,3] [,4]
 [1,] 0.7 0.4 0.1 -0.2
 [2,] 0.4 0.3 0.2 0.1
 [3,] 0.1 0.2 0.3 0.4
 [4,] -0.2 0.1 0.4 0.7
14  矩阵Kronecker积
   n×m矩阵A与h×k矩阵B的kronecker积为一个nh×mk维矩阵,
 在R中kronecker积可以用函数kronecker()来计算,例如:
> A=matrix(1:4,2,2)
 > B=matrix(rep(1,4),2,2)
 > A
   [,1] [,2]
 [1,]   1   3
 [2,]   2   4
 > B
   [,1] [,2]
 [1,]   1   1
 [2,]   1   1
 > kronecker(A,B)
   [,1] [,2] [,3] [,4]
 [1,]   1   1   3   3
 [2,]   1   1   3   3
 [3,]   2   2   4   4
 [4,]   2   2   4   4
15  
   在R中很容易得到一个矩阵的维数,函数dim()将返回一个矩阵的维数,nrow()返回行数,ncol()返回列数,例如:
   > A=matrix(1:12,3,4)
 > A
   [,1] [,2] [,3] [,4]
 [1,]   1   4   7   10
 [2,]   2   5   8   11
 [3,]   3   6   9   12
 > nrow(A)
 [1] 3
 > ncol(A)
 [1] 4
16  
   在R中很容易求得一个矩阵的各行的和、平均数与列的和、平均数,例如:
   > A
   [,1] [,2] [,3] [,4]
 [1,]   1   4   7   10
 [2,]   2   5   8   11
 [3,]   3   6   9   12
 > rowSums(A)
 [1] 22 26 30
 > rowMeans(A)
 [1] 5.5 6.5 7.5
 > colSums(A)
 [1] 6 15 24 33
 > colMeans(A)
 [1] 2 5 8 11
上述关于矩阵行和列的操作,还可以使用apply()函数实现。
> args(apply)
 function (X, MARGIN, FUN, ...)
 其中:x为矩阵,MARGIN用来指定是对行运算还是对列运算,MARGIN=1表示对行运算,MARGIN=2表示对列运算,FUN用来指定运算函数, ...用来给定FUN中需要的其它的参数,例如:
> apply(A,1,sum)
 [1] 22 26 30
 > apply(A,1,mean)
 [1] 5.5 6.5 7.5
 > apply(A,2,sum)
 [1] 6 15 24 33
 > apply(A,2,mean)
 [1] 2 5 8 11
apply()函数功能强大,我们可以对矩阵的行或者列进行其它运算,例如:
 计算每一列的方差
> A=matrix(rnorm(100),20,5)
 > apply(A,2,var)
 [1] 0.4641787 1.4331070 0.3186012 1.3042711 0.5238485
 > apply(A,2,function(x,a)x*a,a=2)
   [,1] [,2] [,3] [,4]
 [1,]   2   8   14   20
 [2,]   4   10   16   22
 [3,]   6   12   18   24
注意:apply(A,2,function(x,a)x*a,a=2)与A*2效果相同,此处旨在说明如何应用alpply函数。
17  
   在统计计算中,我们常常需要计算这样矩阵的逆,如OLS估计中求系数矩阵。R中的包“strucchange”提供了有效的计算方法。
   > args(solveCrossprod)
 function (X, method = c("qr", "chol", "solve"))
其中:method指定求逆方法,选用“qr”效率最高,选用“chol”精度最高,选用“slove”与slove(crossprod(x,x))效果相同,例如:
> A=matrix(rnorm(16),4,4)
 > solveCrossprod(A,method="qr")
       [,1]     [,2]     [,3]     [,4]
 [1,] 0.6132102 -0.1543924 -0.2900796 0.2054730
 [2,] -0.1543924 0.4779277 0.1859490 -0.2097302
 [3,] -0.2900796 0.1859490 0.6931232 -0.3162961
 [4,] 0.2054730 -0.2097302 -0.3162961 0.3447627
 > solveCrossprod(A,method="chol")
       [,1]     [,2]     [,3]     [,4]
 [1,] 0.6132102 -0.1543924 -0.2900796 0.2054730
 [2,] -0.1543924 0.4779277 0.1859490 -0.2097302
 [3,] -0.2900796 0.1859490 0.6931232 -0.3162961
 [4,] 0.2054730 -0.2097302 -0.3162961 0.3447627
 > solveCrossprod(A,method="solve")
       [,1]     [,2]     [,3]     [,4]
 [1,] 0.6132102 -0.1543924 -0.2900796 0.2054730
 [2,] -0.1543924 0.4779277 0.1859490 -0.2097302
 [3,] -0.2900796 0.1859490 0.6931232 -0.3162961
 [4,] 0.2054730 -0.2097302 -0.3162961 0.3447627
 > solve(crossprod(A,A))
       [,1]     [,2]     [,3]     [,4]
 [1,] 0.6132102 -0.1543924 -0.2900796 0.2054730
 [2,] -0.1543924 0.4779277 0.1859490 -0.2097302
 [3,] -0.2900796 0.1859490 0.6931232 -0.3162961
 [4,] 0.2054730 -0.2097302 -0.3162961 0.3447627
18  
   在R中,我们可以很方便的取到一个矩阵的上、下三角部分的元素,函数lower.tri()和函数upper.tri()提供了有效的方法。
   > args(lower.tri)
 function (x, diag = FALSE)
函数将返回一个逻辑值矩阵,其中下三角部分为真,上三角部分为假,选项diag为真时包含对角元素,为假时不包含对角元素。upper.tri()的效果与之孑然相反。例如:
> A
   [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   2   6   10   14
 [3,]   3   7   11   15
 [4,]   4   8   12   16
 > lower.tri(A)
     [,1] [,2] [,3] [,4]
 [1,] FALSE FALSE FALSE FALSE
 [2,] TRUE FALSE FALSE FALSE
 [3,] TRUE TRUE FALSE FALSE
 [4,] TRUE TRUE TRUE FALSE
 > lower.tri(A,diag=T)
   [,1] [,2] [,3] [,4]
 [1,] TRUE FALSE FALSE FALSE
 [2,] TRUE TRUE FALSE FALSE
 [3,] TRUE TRUE TRUE FALSE
 [4,] TRUE TRUE TRUE TRUE
 > upper.tri(A)
     [,1] [,2] [,3] [,4]
 [1,] FALSE TRUE TRUE TRUE
 [2,] FALSE FALSE TRUE TRUE
 [3,] FALSE FALSE FALSE TRUE
 [4,] FALSE FALSE FALSE FALSE
 > upper.tri(A,diag=T)
     [,1] [,2] [,3] [,4]
 [1,] TRUE TRUE TRUE TRUE
 [2,] FALSE TRUE TRUE TRUE
 [3,] FALSE FALSE TRUE TRUE
 [4,] FALSE FALSE FALSE TRUE
 > A[lower.tri(A)]=0
 > A
   [,1] [,2] [,3] [,4]
 [1,]   1   5   9   13
 [2,]   0   6   10   14
 [3,]   0   0   11   15
 [4,]   0   0   0   16
 > A[upper.tri(A)]=0
 > A
   [,1] [,2] [,3] [,4]
 [1,]   1   0   0   0
 [2,]   2   6   0   0
 [3,]   3   7   11   0
 [4,]   4   8   12  
19   backsolve&fowardsolve函数
 这两个函数用于解特殊线性方程组,其特殊之处在于系数矩阵为上或下三角。
> args(backsolve)
 function (r, x, k = ncol(r), upper.tri = TRUE, transpose = FALSE)
 > args(forwardsolve)
 function (l, x, k = ncol(l), upper.tri = FALSE, transpose = FALSE)
其中:r或者l为n×n维三角矩阵,x为n×1维向量,对给定不同的upper.tri和transpose的值,方程的形式不同
 对于函数backsolve()而言,
 例如:
   > A=matrix(1:9,3,3)
 > A
   [,1] [,2] [,3]
 [1,]   1   4   7
 [2,]   2   5   8
 [3,]   3   6   9
 > x=c(1,2,3)
 > x
 [1] 1 2 3
 > B=A
 > B[upper.tri(B)]=0
 > B
   [,1] [,2] [,3]
 [1,]   1   0   0
 [2,]   2   5   0
 [3,]   3   6   9
 > C=A
 > C[lower.tri(C)]=0
 > C
   [,1] [,2] [,3]
 [1,]   1   4   7
 [2,]   0   5   8
 [3,]   0   0   9
 > backsolve(A,x,upper.tri=T,transpose=T)
 [1] 1.00000000 -0.40000000 -0.08888889
 > solve(t(C),x)
 [1] 1.00000000 -0.40000000 -0.08888889
 > backsolve(A,x,upper.tri=T,transpose=F)
 [1] -0.8000000 -0.1333333 0.3333333
 > solve(C,x)
 [1] -0.8000000 -0.1333333 0.3333333
 > backsolve(A,x,upper.tri=F,transpose=T)
 [1] 1.111307e-17 2.220446e-17 3.333333e-01
 > solve(t(B),x)
 [1] 1.110223e-17 2.220446e-17 3.333333e-01
 > backsolve(A,x,upper.tri=F,transpose=F)
 [1] 1 0 0
 > solve(B,x)
 [1] 1.000000e+00 -1.540744e-33 -1.850372e-17
 对于函数forwardsolve()而言,
 例如:
   > A
       [,1] [,2] [,3]
 [1,]   1   4   7
 [2,]   2   5   8
 [3,]   3   6   9
 > B
   [,1] [,2] [,3]
 [1,]   1   0   0
 [2,]   2   5   0
 [3,]   3   6   9
 > C
   [,1] [,2] [,3]
 [1,]   1   4   7
 [2,]   0   5   8
 [3,]   0   0   9
 > x
 [1] 1 2 3
 > forwardsolve(A,x,upper.tri=T,transpose=T)
 [1] 1.00000000 -0.40000000 -0.08888889
 > solve(t(C),x)
 [1] 1.00000000 -0.40000000 -0.08888889
 > forwardsolve(A,x,upper.tri=T,transpose=F)
 [1] -0.8000000 -0.1333333 0.3333333
 > solve(C,x)
 [1] -0.8000000 -0.1333333 0.3333333
 > forwardsolve(A,x,upper.tri=F,transpose=T)
 [1] 1.111307e-17 2.220446e-17 3.333333e-01
 > solve(t(B),x)
 [1] 1.110223e-17 2.220446e-17 3.333333e-01
 > forwardsolve(A,x,upper.tri=F,transpose=F)
 [1] 1 0 0
 > solve(B,x)
 [1] 1.000000e+00 -1.540744e-33 -1.850372e-17
20   row()与col()函数
 在R中定义了的这两个函数用于取矩阵元素的行或列下标矩阵,例如矩阵A={aij}m×n,
 row()函数将返回一个与矩阵A有相同维数的矩阵,该矩阵的第i行第j列元素为i,函数col()类似。例如:
> x=matrix(1:12,3,4)
 > row(x)
   [,1] [,2] [,3] [,4]
 [1,]   1   1   1   1
 [2,]   2   2   2   2
 [3,]   3   3   3   3
 > col(x)
   [,1] [,2] [,3] [,4]
 [1,]   1   2   3   4
 [2,]   1   2   3   4
 [3,]   1   2   3   4
这两个函数同样可以用于取一个矩阵的上下三角矩阵,例如:
 >x
   [,1] [,2] [,3] [,4]
 [1,]   1   4   7   10
 [2,]   2   5   8   11
 [3,]   3   6   9   12
 > x[row(x) 
   
 > x 
   
   
    [,1] [,2] [,3] [,4] 
   
 [1,]   
    1   
    0   
    0   
    0 
   
 [2,]   
    2   
    5   
    0   
    0 
   
 [3,]   
    3   
    6   
    9   
    0 
   
 > x=matrix(1:12,3,4) 
   
 > x[row(x)>col(x)]=0 
   
 > x 
   
   
    [,1] [,2] [,3] [,4] 
   
 [1,]   
    1   
    4   
    7   
    10 
   
 [2,]   
    0   
    5   
    8   
    11 
   
 [3,]   
    0   
    0   
    9   
    12 
   
21   
   
 在R中,函数det(x)将计算方阵x的行列式的值,例如: 
   
> x=matrix(rnorm(16),4,4)
 > x
       [,1]     [,2]     [,3]     [,4]
 [1,] -1.0736375 0.2809563 -1.5796854 0.51810378
 [2,] -1.6229898 -0.4175977 1.2038194 -0.06394986
 [3,] -0.3989073 -0.8368334 -0.6374909 -0.23657088
 [4,] 1.9413061 0.8338065 -1.5877162 -1.30568465
 > det(x)
 [1] 5.717667 
   
22向量化算子
 在R中可以很容易的实现向量化算子,例如: 
   vec<-function (x){
           t(t(as.vector(x)))
 }
 vech<-function (x){
           t(x[lower.tri(x,diag=T)])
 }
 > x=matrix(1:12,3,4)
 > x
   [,1] [,2] [,3] [,4]
 [1,]   1   4   7   10
 [2,]   2   5   8   11
 [3,]   3   6   9   12
 > vec(x)
     [,1]
 [1,]   1
 [2,]   2
 [3,]   3
 [4,]   4
 [5,]   5
 [6,]   6
 [7,]   7
 [8,]   8
 [9,]   9
 [10,]   10
 [11,]   11
 [12,]   12
 > vech(x)
   [,1] [,2] [,3] [,4] [,5] [,6]
 [1,]   1   2   3   5   6   9
23  
   在时间序列分析中,我们常常要用到一个序列的滞后序列,R中的包“fMultivar”中的函数tslag()提供了这个功能。
  > args(tslag)
 function (x, k = 1, trim = FALSE)
其中:x为一个向量,k指定滞后阶数,可以是一个自然数列,若trim为假,则返回序列与原序列长度相同,但含有NA值;若trim项为真,则返回序列中不含有NA值,例如:
> x=1:20
 > tslag(x,1:4,trim=F)
     [,1] [,2] [,3] [,4]
 [1,]   NA   NA   NA   NA
 [2,]   1   NA   NA   NA
 [3,]   2   1   NA   NA
 [4,]   3   2   1   NA
 [5,]   4   3   2   1
 [6,]   5   4   3   2
 [7,]   6   5   4   3
 [8,]   7   6   5   4
 [9,]   8   7   6   5
 [10,]   9   8   7   6
 [11,]   10   9   8   7
 [12,]   11   10   9   8
 [13,]   12   11   10   9
 [14,]   13   12   11   10
 [15,]   14   13   12   11
 [16,]   15   14   13   12
 [17,]   16   15   14   13
 [18,]   17   16   15   14
 [19,]   18   17   16   15
 [20,]   19   18   17   16
 > tslag(x,1:4,trim=T)
     [,1] [,2] [,3] [,4]
 [1,]   4   3   2   1
 [2,]   5   4   3   2
 [3,]   6   5   4   3
 [4,]   7   6   5   4
 [5,]   8   7   6   5
 [6,]   9   8   7   6
 [7,]   10   9   8   7
 [8,]   11   10   9   8
 [9,]   12   11   10   9
 [10,]   13   12   11   10
 [11,]   14   13   12   11
 [12,]   15   14   13   12
 [13,]   16   15   14   13
 [14,]   17   16   15   14
 [15,]   18   17   16   15
 [16,]   19   18   17