Row i is ai column j is aj if u and v are vectors of

Info icon This preview shows pages 14–27. Sign up to view the full content.

View Full Document Right Arrow Icon
Row i is A[i,] , column j is A[,j]. If u and v are vectors of indices (in the correct range), then A[u,v] is the submatrix with rows in u and columns in v . 14
Image of page 14

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
> A<-diag(1:6) > A [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 0 0 0 0 0 [2,] 0 2 0 0 0 0 [3,] 0 0 3 0 0 0 [4,] 0 0 0 4 0 0 [5,] 0 0 0 0 5 0 [6,] 0 0 0 0 0 6 > A[3:1,3:4] [,1] [,2] [1,] 3 0 [2,] 0 0 [3,] 0 0 > A[6,] [1] 0 0 0 0 0 6 15
Image of page 15
2. Basic Matrix Operations 16
Image of page 16

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
If A is a matrix with n rows and m columns, then t (A) is the transposed matrix with m rows and n columns. In matrix notation we write A' . It is clear that A= t ( t (A)) . > x<-matrix(1:6,3,2) > matrix(x,2,3,byrow=TRUE) [,1] [,2] [,3] [1,] 1 2 3 [2,] 4 5 6 > t(x) [,1] [,2] [,3] [1,] 1 2 3 [2,] 4 5 6 17
Image of page 17
Addition Two matrices A and B with the same dimension attribute (the same number of rows and columns) can be added. Actually in R something more general seems to be true. If we want to add vector A to a matrix B , where length(A)<=prod(dim(B)) , then R does dim(A)<-dim(B) before doing A+B . If A is too short, its elements get reused . This still implies only matrices of the same order get added, of course. 18
Image of page 18

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Reusing is useful for adding the same number to all elements of a matrix. > 1+matrix(1:6,3,2) [,1] [,2] [1,] 2 5 [2,] 3 6 [3,] 4 7 Or for adding a vector to all columns of a matrix. > 1:3+matrix(1:6,3,2) [,1] [,2] [1,] 2 5 [2,] 4 7 [3,] 6 9 19
Image of page 19
Other Elementwise Operations Suppose is any binary operator such as addition , subtraction , multiplication , or division . If two matrices A and B have the same dimension, then C=A ‡ B is defined, and C has elements equal to C[i,j]=A[i,j]‡B[i,j] . If is a unary operator such as sin , log , abs , or pnorm then B=‡A is defined and B has elements B[i,j]=‡A[i,j] . 20
Image of page 20

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Vector-vector multiplication In R the inner product of two vectors a and b is sum (a*b) . Remember, the inner product of two vectors is defined by 21 a b = n i =1 a i b i In R the inner product can also be written as x %*% x , but this is pretty ugly, because it is ambiguous (it might as well mean outer (x,x) ). Better is crossprod (x) .
Image of page 21
If x is a vector, then crossprod (x) is i.e. the inner product is the sum of squares. Thus crossprod(x-mean(x))/length(x) is the variance and crossprod(x-mean(x),y-mean(y))/length(x) is the covariance. 22 n i =1 x 2 i
Image of page 22

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
cor<-function(x,y) { nx<-length(x); ny<-length(y) if (nx != ny) stop("not equal length") mx<-mean(x); my<-mean(y) xx<-x-mean(x); yy<-y-mean(y) cxx<-crossprod(xx); cyy<-crossprod(yy) cxy<-crossprod(xx,yy) return(list(n=nx,mx=mx,my=my, vx=cxx/(nx-1),vy=cyy/(ny-1), cxy=cxy/(nx-1), rxy=cxy/sqrt(cxx*cyy))) } 23
Image of page 23
Generalized Outer Product outer() is a very versatile function that can be used to make matrices out of vectors. The simplest use is illustrated below. > outer(1:3,1:2) [,1] [,2] [1,] 1 2 [2,] 2 4 [3,] 3 6 Thus outer( x,y ) gives a matrix A with elements A[i,j]=x[i]*y[j]. 24
Image of page 24

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
> outer(1:3,1:3,"+") [,1] [,2] [,3] [1,] 2 3 4 [2,] 3 4 5 [3,] 4 5 6 > outer(1:3,1:3,"^") [,1] [,2] [,3] [1,] 1 1 1 [2,] 2 4 8 [3,] 3 9 27 > outer(1:3,2:3,function(x,y) x%%y) [,1] [,2] [1,] 1 1 [2,] 0 2 [3,] 1 0 25
Image of page 25
Matrix-Vector Multiplication If A is a matrix with n rows and m columns, then A can be multiplied by a vector x with m elements using A %*% x . The result is a linear combination (weighted sum) of the columns of A , with weights given by x . Thus In the same way x %*% A makes a weighted sum of the rows (thus x must have n elements).
Image of page 26

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 27
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern