http://bbs.sciencenet.cn/home.php?mod=space&uid=443073&do=blog&id=321347
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 创建一个向量
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 创建一个矩阵
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 矩阵转置
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 24
6 矩阵相乘
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,] 90 210 330
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 矩阵对角元素相关运算
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 16
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 1
8 矩阵求逆
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]5111vectors
[,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.eigenvectorsvalues)%*%t(A.eigenvectors) [,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.eigenvectors)%*%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 矩阵的Choleskey分解
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 矩阵奇异值分解
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+011.640705e+003.627301e−16u
[,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,]−0.19033085−0.50893250.42405898[3,]−0.30329950−0.2982646−0.45330031[4,]−0.41626816−0.0875968−0.01637004[5,]−0.529236820.12307110.64231130[6,]−0.642205480.3337389−0.40751869>A.svd=svd(A)>A.svdu%*%diag(A.svdd)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.svdu)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.svdv)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,]0.5477226−0.37816962.641083e−152.056562e−15[4,]0.7302967−0.91247448.583032e−01−2.111449e−16rank
[1] 2
qraux[1]1.182574e+001.156135e+001.513143e+002.111449e−16pivot
[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 矩阵广义逆(Moore-Penrose)
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 16
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 矩阵的维数
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 矩阵的行和、列和、行平均与列平均
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 矩阵X'X的逆
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 取矩阵的上、下三角部分
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 16
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)<col(x)]=0
> 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 行列式的值
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：
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 创建一个矩阵
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 矩阵转置
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 24
6 矩阵相乘
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,] 90 210 330
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 矩阵对角元素相关运算
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 16
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 1
8 矩阵求逆
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]5111vectors
[,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.eigenvectorsvalues)%*%t(A.eigenvectors) [,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.eigenvectors)%*%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 矩阵的Choleskey分解
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 矩阵奇异值分解
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+011.640705e+003.627301e−16u
[,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,]−0.19033085−0.50893250.42405898[3,]−0.30329950−0.2982646−0.45330031[4,]−0.41626816−0.0875968−0.01637004[5,]−0.529236820.12307110.64231130[6,]−0.642205480.3337389−0.40751869>A.svd=svd(A)>A.svdu%*%diag(A.svdd)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.svdu)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.svdv)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,]0.5477226−0.37816962.641083e−152.056562e−15[4,]0.7302967−0.91247448.583032e−01−2.111449e−16rank
[1] 2
qraux[1]1.182574e+001.156135e+001.513143e+002.111449e−16pivot
[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 矩阵广义逆(Moore-Penrose)
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 16
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 矩阵的维数
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 矩阵的行和、列和、行平均与列平均
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 矩阵X'X的逆
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 取矩阵的上、下三角部分
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 16
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)<col(x)]=0
> 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 行列式的值
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 时间序列的滞后值
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 16
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 时间序列的滞后值
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 16
