在R中矩阵相乘使用%*%运算符号.
矩阵相乘必须满足 : 左边的矩阵列数和右边的矩阵行数相等.
例如m行,n列的矩阵和n行p列的矩阵可以相乘, 并且得到的是m行,p列的矩阵.
那么结果矩阵上的每个点的数据是怎么计算来的呢?
例如 :
[2,3] 即第二行第三列这个点.
计算方法 : 左边矩阵的第2行和
右边矩阵的第3列
的每个元素相乘, 然后把所有乘积相加的结果,
例子 :
验证 :
向量也可以和矩阵相乘, 但是同样需要满足条件.
例子2 :
R提供了crossprod和tcrossprod函数, 也是矩阵相乘, 但是速度比%*%更快.(帮助中提到的)
行列变换使用 t() 函数.
crossprod, tcrossprod的y可以不输入.
可以看到:
[参考]
1.
http://blog.csdn.net/myan/article/details/647511
4.
http://f.dataguru.cn/thread-367824-1-1.html
> x <- matrix(data = c(1:6), nrow = 2, ncol = 3, byrow = TRUE)
> x
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
> y <- matrix(data = c(1:12), nrow = 6, ncol = 2, byrow = TRUE)
> y
[,1] [,2]
[1,] 1 2
[2,] 3 4
[3,] 5 6
[4,] 7 8
[5,] 9 10
[6,] 11 12
> y %*% x
[,1] [,2] [,3]
[1,] 9 12 15
[2,] 19 26 33
[3,] 29 40 51
[4,] 39 54 69
[5,] 49 68 87
[6,] 59 82 105验证 :
> z <- y %*% x
> z[3,2]
[1] 40
> sum(y[3,] * x[,2])
[1] 40
> z[6,3]
[1] 105
> sum(y[6,] * x[,3])
[1] 105
> x[,3]
[1] 3 6
> y[6,]
[1] 11 12
3*11+6*12 = 105向量也可以和矩阵相乘, 但是同样需要满足条件.
例如 :
v是向量, a是矩阵,
v %*% a, 那么v的长度必须等于a的行数
. 并且相乘时v转换为(1行多列)的矩阵
反过来,
a %*% v,
那么a的列数必须等于v的长度. 并且相乘时v转换为(多行1列)的矩阵
例子 :
> a <- c(17,19)
> x
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
> a
[1] 17 19
如果左边矩阵的列宽和右边向量的长度不一致, 则无法计算.
反过来, 把向量放在左边, 矩阵放在右边, 那么必须满足左边的向量长度和右边矩阵的行宽要一致.
例子1 :
> x %*% a
Error in x %*% a : non-conformable arguments
以下, 左边向量的长度
等于右边矩阵的行宽
(向量转换为1行2列矩阵)
,
相当于转换为1行2列乘以2行3列.
[1,2] %*% [2,3] ,
得到的是1行3列的矩阵.
> a %*% x
[,1] [,2] [,3]
[1,] 93 129 165
验证
93 = 1*17 + 4*19
129 = 2*17 + 5*19
165 = 3*17 + 6*19例子2 :
> b <- c(100,200,300)
> b %*% x
Error in b %*% x : non-conformable arguments
以下, 左边矩阵的列宽等于右边向量的长度(向量转换为3行1列矩阵).
相当于转换为2行3列乘以3行1列. [2,3] %*% [3,1] , 得到的是2行1列的矩阵.
> x %*% b
[,1]
[1,] 1400
[2,] 3200
验证
1400 = 100*1 + 200*2 + 300*3
3200 = 100*4 + 200*5 + 300*6R提供了crossprod和tcrossprod函数, 也是矩阵相乘, 但是速度比%*%更快.(帮助中提到的)
> help(crossprod)
crossprod package:base R Documentation
Matrix Crossproduct
Description:
Given matrices ‘x’ and ‘y’ as arguments, return a matrix
cross-product. This is formally equivalent to (but usually
slightly faster than) the call ‘t(x) %*% y’ (‘crossprod’) or ‘x
%*% t(y)’ (‘tcrossprod’).
Usage:
crossprod(x, y = NULL)
tcrossprod(x, y = NULL)
Arguments:
x, y: numeric or complex matrices: ‘y = NULL’ is taken to be the
same matrix as ‘x’. Vectors are promoted to single-column or
single-row matrices, depending on the context.
Value:
A double or complex matrix, with appropriate ‘dimnames’ taken from
‘x’ and ‘y’.
Note:
When ‘x’ or ‘y’ are not matrices, they are treated as column or
row matrices, but their ‘names’ are usually *not* promoted to
‘dimnames’. Hence, currently, the last example has empty
dimnames.
See Also:
‘%*%’ and outer product ‘%o%’.
Examples:
(z <- crossprod(1:4)) # = sum(1 + 2^2 + 3^2 + 4^2)
drop(z) # scalar
x <- 1:4; names(x) <- letters[1:4]; x
tcrossprod(as.matrix(x)) # is
identical(tcrossprod(as.matrix(x)),
crossprod(t(x)))
tcrossprod(x) # no dimnames
m <- matrix(1:6, 2,3) ; v <- 1:3; v2 <- 2:1
stopifnot(identical(tcrossprod(v, m), v %*% t(m)),
identical(tcrossprod(v, m), crossprod(v, t(m))),
identical(crossprod(m, v2), t(m) %*% v2))行列变换使用 t() 函数.
> 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
> t(x)
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9
[4,] 10 11 12
> x %*% t(x)
[,1] [,2] [,3]
[1,] 166 188 210
[2,] 188 214 240
[3,] 210 240 270
> t(x) %*% x
[,1] [,2] [,3] [,4]
[1,] 14 32 50 68
[2,] 32 77 122 167
[3,] 50 122 194 266
[4,] 68 167 266 365
> crossprod(x,x)
[,1] [,2] [,3] [,4]
[1,] 14 32 50 68
[2,] 32 77 122 167
[3,] 50 122 194 266
[4,] 68 167 266 365
> tcrossprod(x,x)
[,1] [,2] [,3]
[1,] 166 188 210
[2,] 188 214 240
[3,] 210 240 270crossprod, tcrossprod的y可以不输入.
> crossprod(x)
[,1] [,2] [,3] [,4]
[1,] 14 32 50 68
[2,] 32 77 122 167
[3,] 50 122 194 266
[4,] 68 167 266 365
> tcrossprod(x)
[,1] [,2] [,3]
[1,] 166 188 210
[2,] 188 214 240
[3,] 210 240 270可以看到:
> crossprod(x,x) == t(x) %*% x
[,1] [,2] [,3] [,4]
[1,] TRUE TRUE TRUE TRUE
[2,] TRUE TRUE TRUE TRUE
[3,] TRUE TRUE TRUE TRUE
[4,] TRUE TRUE TRUE TRUE
> tcrossprod(x,x) == x %*% t(x)
[,1] [,2] [,3]
[1,] TRUE TRUE TRUE
[2,] TRUE TRUE TRUE
[3,] TRUE TRUE TRUE> help(t)
t package:base R Documentation
Matrix Transpose
Description:
Given a matrix or ‘data.frame’ ‘x’, ‘t’ returns the transpose of
‘x’.
Usage:
t(x)
Arguments:
x: a matrix or data frame, typically.
Details:
This is a generic function for which methods can be written. The
description here applies to the default and ‘"data.frame"’
methods.
A data frame is first coerced to a matrix: see ‘as.matrix’. When
‘x’ is a vector, it is treated as a column, i.e., the result is a
1-row matrix.
Value:
A matrix, with ‘dim’ and ‘dimnames’ constructed appropriately from
those of ‘x’, and other attributes except names copied across.[参考]
5. R
> help("%*%")
matmult package:base R Documentation
Matrix Multiplication
Description:
Multiplies two matrices, if they are conformable. If one argument
is a vector, it will be promoted to either a row or column matrix
to make the two arguments conformable. If both are vectors it
will return the inner product (as a matrix).
Usage:
x %*% y
Arguments:
x, y: numeric or complex matrices or vectors.
Details:
When a vector is promoted to a matrix, its names are not promoted
to row or column names, unlike ‘as.matrix’.
This operator is S4 generic but not S3 generic. S4 methods need
to be written for a function of two arguments named ‘x’ and ‘y’.
Value:
A double or complex matrix product. Use ‘drop’ to remove
dimensions which have only one level.
References:
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S
Language_. Wadsworth & Brooks/Cole.
See Also:
‘matrix’, ‘Arithmetic’, ‘diag’.
Examples:
x <- 1:4
(z <- x %*% x) # scalar ("inner") product (1 x 1 matrix)
drop(z) # as scalar
y <- diag(x)
z <- matrix(1:12, ncol = 3, nrow = 4)
y %*% z
y %*% x
x %*% z
