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

Matrix Algebra Review

Create Matrix Object

Assign Dimension Names

Subset Elements of a Matrix

Subset Elements of a Matrix

Create Vector Object

Coerce Vector to Matrix

Transpose of Matrix

Symmetric Matrix

Basic Matrix Operations (element by element)

Matrix Addition and Subtraction

Scalar Multiplication

Matrix Multiplication

Matrix Multiplication (not element by element)

Identity Matrix

Diagonal Matrices

Matrix Inverse

Solving Systems of Linear Equations

Solving Systems of Linear Equations

Summation Using Matrix Algebra

Sum of Squares

Sum of Cross Products