26 ax a 1 x 1 a 2 x 2 a m x m a 1 2 1 090 0077 2 076

Info icon This preview shows pages 26–39. Sign up to view the full content.

View Full Document Right Arrow Icon
26 Ax = A [ , 1] * x [1] + A [ , 2] * x [2] + · · · + A [ , m ] * x [ m ]
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
> a [,1] [,2] [1,] 0.90 0.077 [2,] -0.76 -0.922 [3,] 0.37 -1.115 [4,] -0.17 0.450 [5,] -0.70 1.063 > b<-1:2 > a%*%b [,1] [1,] 1.05 [2,] -2.60 [3,] -1.86 [4,] 0.73 [5,] 1.43 > drop(a%*%b) [1] 1.05 -2.60 -1.86 0.73 1.43 27
Image of page 27
Observe that the result is a matrix. If you use the %*% symbol, for matrix multiplication, then R first makes the vector into an m x 1 matrix. In the same way (remember b is c (1,2) ) > b%*%b [,1] [1,] 5 The drop() function, applied to an array, drops all dimensions of the array with value equal to one, so it makes a vector from an m x 1 matrix. You could also use as.vector() . 28
Image of page 28

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

View Full Document Right Arrow Icon
Matrix-Matrix Multiplication If A is a matrix with dim(A)=c(n,m) and B is a matrix with dim(B)=c(m,r) , then C=A %*% B is a matrix with dim(C)=c(n,r) . Thus the number of columns of A has to be equal to the number of rows of B (the matrices must conform ). The columns are the vectors being combined in the weighted sum, the rows are the coefficients (weights). 29
Image of page 29
Matrix multiplication AB creates r weighted sums of the m columns of A (or n weighted sums of the rows of B ). In summation notation C=AB can be written out as In terms of inner products the matrix multiplication C=AB can be written as C[i,j]= sum (A[i,]*B[,j]) . 30 c ij = m =1 a i b j
Image of page 30

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

View Full Document Right Arrow Icon
More generally 31 C = AB c ij = m =1 a i b j , C = AB c ij = m =1 a i b j , C = A B c ij = m =1 a i b j , C = A B c ij = m =1 a i b j .
Image of page 31
Observe that t(A %*% B)=t(B) %*% t(A) , or because In general, if A and B are matrices, then crossprod (A,B) is t(A) %*% B , while tcrossprod (A,B) is A %*%t (B) . Thus for vectors tcrossprod (x,y) is outer (x,y) . If crossprod () or tcrossprod () only have a single argument, they compute their own cross products, so crossprod (X) is t (X)%*% X , and tcrossprod (X) is X %*%t (X) . 32 [( AB ) ] ij = ( AB ) ji = m =1 a j b i = ( B A ) ij ( AB ) = B A
Image of page 32

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

View Full Document Right Arrow Icon
It is important to realize that matrix multiplication is distributive because A(B+C) =AB+AC and (B+C)A=BA+CA , but not necessarily commutative . In general AB BA . In fact if AB is defined, BA may not even be defined because the matrices do not conform. But even for square matrices of the same order, which always conform, we generally have AB BA . 33
Image of page 33
Trace The trace of a square matrix A is the sum of its diagonal elements. Of course for any matrix X both crossprod (X) and tcrossprod (X) are square, and thus we can enquire about the trace of crossprod (X) and tcrossprod (X) . Turns out that 34 tr X X = tr XX = n i =1 m j =1 x 2 ij
Image of page 34

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

View Full Document Right Arrow Icon
More generally Also very useful later on 35 tr ABC = tr BCA = tr CAB. tr A B = m j =1 ( A B ) jj = m j =1 n i =1 ( A ) ji b ij = m j =1 n i =1 a ij b ij
Image of page 35
3. Patterned Matrices 36
Image of page 36

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

View Full Document Right Arrow Icon
A matrix A is square if the number of rows is equal to the number of columns. A rectangular matrix is not (necessarily) square. A n x m matrix is tall if n m and broad if n m . square tall broad 37
Image of page 37
A square matrix A is symmetric if A[i,j]=A[j,i] for all i and j . Equivalently: if A=t(A) . A square matrix is asymmetric if it is not symmetric. A square matrix A is anti - symmetric if A[i,j]=-A[j,i] for all i and j . Equivalently: if t(A)=-A .
Image of page 38

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

View Full Document Right Arrow Icon
Image of page 39
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