3.3 Representing Summation Using Matrix Notation

Consider the sum: \[ \sum_{k=1}^{n}x_{k}=x_{1}+\cdots+x_{n}. \] Let \(\mathbf{x}=(x_{1},\ldots,x_{n})^{\prime}\) be an \(n\times1\) vector and \(\mathbf{1}=(1,\ldots,1)^{\prime}\) be an \(n\times1\) vector of ones. Then, \[ \mathbf{x}^{\prime}\mathbf{1}=\left[\begin{array}{ccc} x_{1} & \ldots & x_{n}\end{array}\right]\cdot\left[\begin{array}{c} 1\\ \vdots\\ 1 \end{array}\right]=x_{1}+\cdots+x_{n}=\sum_{k=1}^{n}x_{k}, \] and, \[ \mathbf{1}^{\prime}\mathbf{x}=\left[\begin{array}{ccc} 1 & \ldots & 1\end{array}\right]\cdot\left[\begin{array}{c} x_{1}\\ \vdots\\ x_{n} \end{array}\right]=x_{1}+\cdots+x_{n}=\sum_{k=1}^{n}x_{k}. \] Next, consider the sum of squared \(x\) values, \[ \sum_{k=1}^{n}x_{k}^{2}=x_{1}^{2}+\cdots+x_{n}^{2}. \] This sum can be conveniently represented as, \[ \mathbf{x}^{\prime}\mathbf{x}=\left[\begin{array}{ccc} x_{1} & \ldots & x_{n}\end{array}\right]\cdot\left[\begin{array}{c} x_{1}\\ \vdots\\ x_{n} \end{array}\right]=x_{1}^{2}+\cdots+x_{n}^{2}=\sum_{k=1}^{n}x_{k}^{2}. \] Last, consider the sum of cross products, \[ \sum_{k=1}^{n}x_{k}y_{k}=x_{1}y_{1}+\cdots x_{n}y_{n}. \] This sum can be compactly represented by, \[ \mathbf{x}^{\prime}\mathbf{y}=\left[\begin{array}{ccc} x_{1} & \ldots & x_{n}:\end{array}\right]\cdot\left[\begin{array}{c} y_{1}\\ \vdots\\ y_{n} \end{array}\right]=x_{1}y_{1}+\cdots x_{n}y_{n}=\sum_{k=1}^{n}x_{k}y_{k}. \] Note that \(\mathbf{x}^{\prime}\mathbf{y=y}^{\prime}\mathbf{x}\).

Example 2.17 (Computing sums in R)

In R, summing the elements in a vector can be done using matrix algebra.

# create vector of 1's and a vector of x
onevec = rep(1,3)
onevec

## [1] 1 1 1

xvec = c(1,2,3)
# sum elements in x
t(xvec)%*%onevec

##      [,1]
## [1,]    6

The functions crossprod() and sum() are generally computationally more efficient:

crossprod(xvec,onevec)

##      [,1]
## [1,]    6

sum(xvec)

## [1] 6

Sums of squares are best computed using:

crossprod(xvec)

##      [,1]
## [1,]   14

sum(xvec^2)

## [1] 14

The dot-product two vectors can be conveniently computed using the crossprod() function:

yvec = 4:6
xvec

## [1] 1 2 3

yvec

## [1] 4 5 6

crossprod(xvec,yvec)

##      [,1]
## [1,]   32

crossprod(yvec,xvec)

##      [,1]
## [1,]   32

\(\blacksquare\)