Create Matrix Object

You create matrices using the matrix() constructor function:

args(matrix)

## function (data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL) 
## NULL

# fill matrix by column (default)
matA = matrix(data=c(1,2,3,4,5,6), nrow=2, ncol=3)
matA

##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6

class(matA)

## [1] "matrix"

# fill matrix by row
matA = matrix(data=c(1,2,3,4,5,6), nrow=2, ncol=3, byrow=TRUE)
matA

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6

Assign Dimension Names

You can name the rows and columns of a matrix using the functions dimnames(), rownames() and colnames():

dim(matA)

## [1] 2 3

dimnames(matA)

## NULL

dimnames(matA) = list(c("row1","row2"),c("col1","col2","col3"))
matA

##      col1 col2 col3
## row1    1    2    3
## row2    4    5    6

colnames(matA) = c("Col1", "Col2", "Col3")
rownames(matA) = c("Row1", "Row2")
matA

##      Col1 Col2 Col3
## Row1    1    2    3
## Row2    4    5    6

Subset Elements of a Matrix

You can subset the elements of a matrix by position or name:

# subset by position
matA[1, 2]

## [1] 2

# subset by name
matA["Row1", "Col1"]

## [1] 1

You can extract entire rows or columns

# extract first row
matA[1, ]

## Col1 Col2 Col3 
##    1    2    3

# extract 2nd column
matA[, 2]

## Row1 Row2 
##    2    5

Subset Elements of a Matrix

When you extract an entire row or column of a matrix, R drops the dimension attributes of the result. That is, what you get is a numeric vector and not a row or column matrix. To preserve the dimension attributes the optional argument drop=FALSE:

# preserve dimension attributes using drop=FALSE
matA[1, , drop=FALSE]

##      Col1 Col2 Col3
## Row1    1    2    3

matA[, 2, drop=FALSE]

##      Col2
## Row1    2
## Row2    5

Create Vector Object

Vectors in R do not have dimension attributes. They appear as row vectors but that is just how they are printed on screen.

xvec = c(1,2,3)
xvec

## [1] 1 2 3

class(xvec)

## [1] "numeric"

dim(xvec)

## NULL

names(xvec) = c("x1", "x2", "x3")
xvec

## x1 x2 x3 
##  1  2  3

Coerce Vector to Matrix

# coerce vector to class matrix: note column vector is created
xvec = as.matrix(xvec)
xvec

##    [,1]
## x1    1
## x2    2
## x3    3

class(xvec)

## [1] "matrix"

# create row vector
xvec = c(1,2,3)
xvec = matrix(xvec, nrow=1)
xvec

##      [,1] [,2] [,3]
## [1,]    1    2    3

Note: R has many as. functions for coercing the class of an object to another class.

Transpose of Matrix

Transposing a matrix interchanges rows and columns:

matA = matrix(data=c(1,2,3,4,5,6), nrow=2, ncol=3, 
              byrow=TRUE)
t(matA)

##      [,1] [,2]
## [1,]    1    4
## [2,]    2    5
## [3,]    3    6

xvec = c(1,2,3)
t(xvec)

##      [,1] [,2] [,3]
## [1,]    1    2    3

Symmetric Matrix

A symmetric matrix is equal to its transpose:

matS = matrix(c(1,2,2,1),2,2)
matS

##      [,1] [,2]
## [1,]    1    2
## [2,]    2    1

# check for symmetry
matS == t(matS)

##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

Basic Matrix Operations (element by element)

Element by element operations require that all matrices have the same dimension and apply to each element the matrices.

matA = matrix(c(4,9,2,1),2,2, byrow=TRUE)
matB = matrix(c(2,0,0,7),2,2, byrow=TRUE)
matA

##      [,1] [,2]
## [1,]    4    9
## [2,]    2    1

matB

##      [,1] [,2]
## [1,]    2    0
## [2,]    0    7

Matrix Addition and Subtraction

Matrix addition and subtraction are element by element operations:

# matrix addition
matC = matA + matB
matC

##      [,1] [,2]
## [1,]    6    9
## [2,]    2    8

# matrix subtraction
matC = matA - matB
matC

##      [,1] [,2]
## [1,]    2    9
## [2,]    2   -6

Scalar Multiplication

Multipling a matrix by a scalar (single number) is an element by element operation:

matA = matrix(c(3,-1,0,5), 2, 2, 
              byrow=TRUE)
matA

##      [,1] [,2]
## [1,]    3   -1
## [2,]    0    5

matC = 2*matA
matC

##      [,1] [,2]
## [1,]    6   -2
## [2,]    0   10

Matrix Multiplication

Matrix multiplication only works on conformable matrices:

matA = matrix(1:4,2,2, byrow=TRUE)
matB = matrix(c(1,2,1,3,4,2),2,3, byrow=TRUE)
matA

##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4

matB

##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    4    2

dim(matA)

## [1] 2 2

dim(matB)

## [1] 2 3

Matrix Multiplication (not element by element)

matC = matA%*%matB
matC

##      [,1] [,2] [,3]
## [1,]    7   10    5
## [2,]   15   22   11

The $(i,j)$ element of matC is the dot product of the i’th row of matA with the j’th column of matB

Multiplying two non-conformable matrices results in an error:

# here B%*%A doesn't work b/c A and B are not comformable
# matB%*%matA
# Error in matB %*% matA : non-conformable arguments

Identity Matrix

Create identity matrix of dimension 2:

matI = diag(2)
matI

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

Multiplying a matrix by the identity matrix returns the original matrix.

matA = matrix(c(1,2,3,4), 2, 2, byrow=TRUE)
matI%*%matA

##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4

matA%*%matI

##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4

Diagonal Matrices

Create a diagonal matrix with $c=(1,2,3)$ along the main diagonal.

matD = diag(1:3)
matD

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    2    0
## [3,]    0    0    3

Matrix Inverse

In R, the function solve() computes the inverse of a matrix.

matA

##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4

matA.inv = solve(matA)
matA.inv

##      [,1] [,2]
## [1,] -2.0  1.0
## [2,]  1.5 -0.5

matA%*%matA.inv

##      [,1]     [,2]
## [1,]    1 1.11e-16
## [2,]    0 1.00e+00

matA.inv%*%matA

##      [,1]     [,2]
## [1,]    1 4.44e-16
## [2,]    0 1.00e+00

Solving Systems of Linear Equations

The solution of two linear equations in two unknows is at the intersection of the two lines:

curve(1-x, 0, 1, lwd=2, ylab="y")
abline(a=-1, b=2, lwd=2)
title("x+y=1, 2x-y=1")

Solving Systems of Linear Equations

The system of linear equations can be expressed in matrix form as $Az = b$. The solution is $z = A^{-1}b$:

matA = matrix(c(1,1,2,-1), 2, 2, byrow=TRUE)
vecB = c(1,1)
matA.inv = solve(matA)
z = matA.inv%*%vecB
z

##       [,1]
## [1,] 0.667
## [2,] 0.333

Summation Using Matrix Algebra

$$\sum_{k=1}^{n}x_{k} = \mathbf{x}^{\prime}\mathbf{1}$$

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

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

# more efficient
crossprod(xvec, onevec)

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

# more efficient still - returns a number
sum(xvec)

## [1] 6

Sum of Squares

$$\sum_{k=1}^{n}x_{k}^{2} = \mathbf{x}^{\prime}\mathbf{x}$$

xvec

## [1] 1 2 3

t(xvec)%*%xvec

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

# more efficient
crossprod(xvec)

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

# most efficient
sum(xvec^2)

## [1] 14

Sum of Cross Products

$$\sum_{k=1}^{n}x_{k}y_{k} = \mathbf{x}^{\prime}\mathbf{y}$$

yvec = 4:6
xvec

## [1] 1 2 3

yvec

## [1] 4 5 6

t(xvec)%*%yvec

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

crossprod(yvec, xvec)

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

sum(xvec*yvec)

## [1] 32