Matrices in R

Scalar Multiplication

#Make a matrix
X <- matrix(1:15, 5, 3)
print(X)

##      [,1] [,2] [,3]
## [1,]    1    6   11
## [2,]    2    7   12
## [3,]    3    8   13
## [4,]    4    9   14
## [5,]    5   10   15

#Multiple matrix by scalar, e.g. 2
a <- 2
print(a*X)

##      [,1] [,2] [,3]
## [1,]    2   12   22
## [2,]    4   14   24
## [3,]    6   16   26
## [4,]    8   18   28
## [5,]   10   20   30

Transpose

Converting rows to columns, and vice versa. This is done very simply in R using the t function on a matrix.

#Make a matrix
X <- matrix(1:15, 5, 3)
print(X)

##      [,1] [,2] [,3]
## [1,]    1    6   11
## [2,]    2    7   12
## [3,]    3    8   13
## [4,]    4    9   14
## [5,]    5   10   15

#Transpose matrix, X
t(X)

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    2    3    4    5
## [2,]    6    7    8    9   10
## [3,]   11   12   13   14   15

Identity Matrix

An identity matrix of \(N \times N\) dimensions can easily be constructed in R using the diag function.

diag(4)

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1

This gives a 4x4 identity matrix.

Inverse

Some, but not all, squared matrices have a corresponding inverse matrix. The inverse matrix (\(X^{-1}\)) is that matrix, which when multiplied by the original matrix (\(X\)), gives the identity matrix, \(I\).

\[X^{-1} \cdot X = I\]

In R, the inverse matrix can be obtained using the solve() function.

Matrix Multiplication

Generally, works so long as the number of columns of the first matrix has the same number of rows as the second matrix in our multiplication equation.

\[m \times n \cdot n \times p = m \times p\]

This highlights the dimensions of the matrices being multiplied and that the \(n\) of each matrix needs to be the same in order for this to work.

Matrix multiplication in R is done using %*%, e.g. x %*% y.

a <- matrix(1:12, nrow=4)
b <- matrix(1:15, nrow=3)

a %*% b

##      [,1] [,2] [,3] [,4] [,5]
## [1,]   38   83  128  173  218
## [2,]   44   98  152  206  260
## [3,]   50  113  176  239  302
## [4,]   56  128  200  272  344

Let’s use the following example to illustrate how this computation can be done in R:

\[ \begin{equation} 2x - y = 0 \\ -x + 2y - z = -1 \\ -3y + 4z = 4 \end{equation} \]

Solving this matrix multiplication involves taking the inverse matrix as seen below: \[ \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -3 & 4 \end{bmatrix}^{-1} \begin{bmatrix} 0 \\ -1 \\ 4 \end{bmatrix} \]

The solution gives us the values of \(x, y,\) and \(z\).

A <- matrix(c(2, -1, 0, -1, 2, -3, 0, -1, 4), 3, 3)

# can also be created using:
# cbind(c(2, -1, 0), c(-1, 2, -3), c(0, -1, 4))

x <- matrix(c(0, -1, 4), 3, 1)

solve(A) %*% x

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

# can also be written as:
# solve(A, x)

Just to prove that solve(A) gives the inverse matrix, \(I\), and that \(I^{-1} \times I\) gives the identity matrix, let’s try it below.

solve(A) # gives the inverse matrix of A

##           [,1]      [,2]      [,3]
## [1,] 0.8333333 0.6666667 0.1666667
## [2,] 0.6666667 1.3333333 0.3333333
## [3,] 0.5000000 1.0000000 0.5000000

solve(A) %*% A  #should give the identiy matrix

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

These notes were compiled from the Rafael Irizarry’s MOOC on the subject.