极客教程R语言 对矩阵进行代数运算
矩阵 是一个以行和列排列的矩形数字。在一个矩阵中，我们知道行是水平方向的，列是垂直方向的。在R中，矩阵是二维的、同质的数据结构。这些是矩阵的一些例子。
基本代数运算是任何一种传统的算术运算，即加法、减法、乘法、除法、提高到整数幂和取根。这些运算可以在数字上进行，在这种情况下，它们通常被称为算术运算。我们可以在R语言中对矩阵进行更多的代数运算。 在R语言中可以对矩阵进行的代数运算。
- 对单个矩阵的操作
- 一元运算
- 二元运算
- 线性代数操作
- 矩阵的秩、行列式、转置、逆置、跟踪
- 矩阵的空洞性
- 矩阵的特征值和特征向量
- 解决一个线性矩阵方程
对单个矩阵的操作
我们可以使用重载算术运算符对矩阵进行逐元运算，以创建一个新矩阵。在+=, -=, *=运算符的情况下，现有的矩阵被修改。
# R program to demonstrate
# basic operations on a single matrix
# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The 3x3 matrix:\n")
print(a)
# add 1 to every element
cat("Adding 1 to every element:\n")
print(a + 1)
# subtract 3 from each element
cat("Subtracting 3 from each element:\n")
print(a-3)
# multiply each element by 10
cat("Multiplying each element by 10:\n")
print(a * 10)
# square each element
cat("Squaring each element:\n")
print(a ^ 2)
# modify existing matrix
cat("Doubled each element of original matrix:\n")
print(a * 2)
输出
The 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 1 2 3
[2, ] 4 5 6
[3, ] 7 8 9
Adding 1 to every element:
[, 1] [, 2] [, 3]
[1, ] 2 3 4
[2, ] 5 6 7
[3, ] 8 9 10
Subtracting 3 from each element:
[, 1] [, 2] [, 3]
[1, ] -2 -1 0
[2, ] 1 2 3
[3, ] 4 5 6
Multiplying each element by 10:
[, 1] [, 2] [, 3]
[1, ] 10 20 30
[2, ] 40 50 60
[3, ] 70 80 90
Squaring each element:
[, 1] [, 2] [, 3]
[1, ] 1 4 9
[2, ] 16 25 36
[3, ] 49 64 81
Doubled each element of original matrix:
[, 1] [, 2] [, 3]
[1, ] 2 4 6
[2, ] 8 10 12
[3, ] 14 16 18
单项操作
在R语言中可以对矩阵进行许多单项操作，包括总和、最小、最大等。
# R program to demonstrate
# unary operations on a matrix
# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The 3x3 matrix:\n")
print(a)
# maximum element in the matrix
cat("Largest element is:\n")
print(max(a))
# minimum element in the matrix
cat("Smallest element is:\n")
print(min(a))
# sum of element in the matrix
cat("Sum of elements is:\n")
print(sum(a))
输出
The 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 1 2 3
[2, ] 4 5 6
[3, ] 7 8 9
Largest element is:
[1] 9
Smallest element is:
[1] 1
Sum of elements is:
[1] 45
二元操作
这些操作以元素方式应用于矩阵，并创建一个新的矩阵。你可以使用所有的基本算术运算符，如+, -, *, /等。如果使用+=, -=, =运算符，现有的矩阵会被修改。
# R program to demonstrate
# binary operations on a matrix
# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The 3x3 matrix:\n")
print(a)
# Create another 3x3 matrix
b = matrix(
c(1, 2, 5, 4, 6, 2, 9, 4, 3),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The another 3x3 matrix:\n")
print(b)
cat("Matrix addition:\n")
print(a + b)
cat("Matrix subtraction:\n")
print(a-b)
cat("Matrix element wise multiplication:\n")
print(a * b)
cat("Regular Matrix multiplication:\n")
print(a %*% b)
cat("Matrix elementwise division:\n")
print(a / b)
输出
The 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 1 2 3
[2, ] 4 5 6
[3, ] 7 8 9
The another 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 1 2 5
[2, ] 4 6 2
[3, ] 9 4 3
Matrix addition:
[, 1] [, 2] [, 3]
[1, ] 2 4 8
[2, ] 8 11 8
[3, ] 16 12 12
Matrix subtraction:
[, 1] [, 2] [, 3]
[1, ] 0 0 -2
[2, ] 0 -1 4
[3, ] -2 4 6
Matrix element wise multiplication:
[, 1] [, 2] [, 3]
[1, ] 1 4 15
[2, ] 16 30 12
[3, ] 63 32 27
Regular Matrix multiplication:
[, 1] [, 2] [, 3]
[1, ] 36 26 18
[2, ] 78 62 48
[3, ] 120 98 78
Matrix elementwise division:
[, 1] [, 2] [, 3]
[1, ] 1.0000000 1.0000000 0.6
[2, ] 1.0000000 0.8333333 3.0
[3, ] 0.7777778 2.0000000 3.0
线性代数操作
人们可以在R中对一个给定的矩阵进行许多线性代数操作，其中一些操作如下。
- 矩阵的秩、行列式、转置、逆、迹。
# R program to demonstrate
# Linear algebraic operations on a matrix
# Importing required library
library(pracma) # For rank of matrix
library(psych) # For trace of matrix
# Create a 3x3 matrix
A = matrix(
c(6, 1, 1, 4, -2, 5, 2, 8, 7),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The 3x3 matrix:\n")
print(A)
# Rank of a matrix
cat("Rank of A:\n")
print(Rank(A))
# Trace of matrix A
cat("Trace of A:\n")
print(tr(A))
# Determinant of a matrix
cat("Determinant of A:\n")
print(det(A))
# Transpose of a matrix
cat("Transpose of A:\n")
print(t(A))
# Inverse of matrix A
cat("Inverse of A:\n")
print(inv(A))
- 输出
The 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 6 1 1
[2, ] 4 -2 5
[3, ] 2 8 7
Rank of A:
[1] 3
Trace of A:
[1] 11
Determinant of A:
[1] -306
Transpose of A:
[, 1] [, 2] [, 3]
[1, ] 6 4 2
[2, ] 1 -2 8
[3, ] 1 5 7
Inverse of A:
[, 1] [, 2] [, 3]
[1, ] 0.17647059 -0.003267974 -0.02287582
[2, ] 0.05882353 -0.130718954 0.08496732
[3, ] -0.11764706 0.150326797 0.05228758
- 矩阵的无效性。
# R program to demonstrate
# nullity of a matrix
# Importing required library
library(pracma)
# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The 3x3 matrix:\n")
print(a)
# No of column
col = ncol(a)
# Rank of matrix
rank = Rank(a)
# Calculating nullity
nullity = col - rank
cat("Nullity of matrix is:\n")
print(nullity)
- 输出
The 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 1 2 3
[2, ] 4 5 6
[3, ] 7 8 9
Nullity of matrix is:
[1] 1
- 矩阵的特征值和特征向量。
# R program to illustrate
# Eigenvalues and eigenvectors of metrics
# Create a 3x3 matrix
A = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9),
nrow = 3,
ncol = 3,
byrow = TRUE
)
cat("The 3x3 matrix:\n")
print(A)
# Calculating Eigenvalues and eigenvectors
print(eigen(A))
- 输出
The 3x3 matrix:
[, 1] [, 2] [, 3]
[1, ] 1 2 3
[2, ] 4 5 6
[3, ] 7 8 9
eigen() decomposition
values
[1] 1.611684e+01 -1.116844e+00 -1.303678e-15vectors
[, 1] [, 2] [, 3]
[1, ] -0.2319707 -0.78583024 0.4082483
[2, ] -0.5253221 -0.08675134 -0.8164966
[3, ] -0.8186735 0.61232756 0.4082483
- 解决一个线性矩阵方程。
# R program to illustrate
# Solve a linear matrix equation of metrics
# Importing library for applying pseudoinverse
library(MASS)
# Create a 2x2 matrix
A = matrix(
c(1, 2, 3, 4),
nrow = 2,
ncol = 2,
)
cat("A = :\n")
print(A)
# Create another 2x1 matrix
b = matrix(
c(7, 10),
nrow = 2,
ncol = 1,
)
cat("b = :\n")
print(b)
cat("Solution of linear equations:\n")
print(solve(A)%*% b)
cat("Solution of linear equations using pseudoinverse:\n")
print(ginv(A)%*% b)
- 输出
A = :
[, 1] [, 2]
[1, ] 1 3
[2, ] 2 4
b = :
[, 1]
[1, ] 7
[2, ] 10
Solution of linear equations:
[, 1]
[1, ] 1
[2, ] 2
Solution of linear equations using pseudoinverse:
[, 1]
[1, ] 1
[2, ] 2