Matrices

Creating matrices

You're all familiar with what a matrix is mathematically. It is a 2-dimensional array of symbols -- like a 2-dimensional vector. In fact, that's how R implements matrices; they are vectors organized into an array with a multidimensional index.

Matrices are instantiated with the `matrix` function, in which we specify the vector of entries and the numbers of rows and columns.

``````v1 <- 1:9  # a vector of length 9 = 3 x 3
m1 <- matrix(v1, nrow = 3, ncol = 3)
m1
``````
``````##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9
``````

You see it filled in the entries going down the column. If you try to form a matrix with a vector that's too short, R pads it with NA's.

Matrices of characters or logicals can also be created.

``````m2 <- matrix(letters[1:8], nrow = 2, ncol = 4)
m2
``````
``````##      [,1] [,2] [,3] [,4]
## [1,] "a"  "c"  "e"  "g"
## [2,] "b"  "d"  "f"  "h"
``````

But if you try to mix numeric and character entries, the numbers are converted to characters, just like with vectors.

Names and dimensions

R has functions that return dimnesions just like `length` works for vectors.

``````nrow(m2)
``````
``````## [1] 2
``````
``````ncol(m2)
``````
``````## [1] 4
``````
``````dim(m2)
``````
``````## [1] 2 4
``````

Just like a vector can have names associated to the entries, a matrix can have names for columns and rows.

``````dim(m1)
``````
``````## [1] 3 3
``````
``````colnames(m1) <- c("C1", "C2", "C3")
rownames(m1) <- c("R1", "R2", "R3")
m1
``````
``````##    C1 C2 C3
## R1  1  4  7
## R2  2  5  8
## R3  3  6  9
``````

Identifying entries rows and columns

There are no surprises in how a cell in a matrix is identified.

``````m2
``````
``````##      [,1] [,2] [,3] [,4]
## [1,] "a"  "c"  "e"  "g"
## [2,] "b"  "d"  "f"  "h"
``````
``````m2[1, 2]  # row 1, column 2
``````
``````## [1] "c"
``````

You can also use row and column names to pin down an entry.

``````m1["R2", "C3"]
``````
``````## [1] 8
``````

Specifying ranges of columns and rows will give a submatrix.

``````m1[1:2, 2:3]  # The row and column names ride along
``````
``````##    C2 C3
## R1  4  7
## R2  5  8
``````

Leaving one slot empty fetches all rows or columns.

``````m2[, 1:3]
``````
``````##      [,1] [,2] [,3]
## [1,] "a"  "c"  "e"
## [2,] "b"  "d"  "f"
``````

Note that if you specify a single row or column, R drops the matrix structure and returns a vector.

``````v2 <- m1[1, ]
v2
``````
``````## C1 C2 C3
##  1  4  7
``````
``````class(v2)
``````
``````## [1] "integer"
``````

Conditionals, logicals and subsetting

Equalities and inequalities of matrices return logical matrices of the same dimensions.

``````m1 > 4
``````
``````##       C1    C2   C3
## R1 FALSE FALSE TRUE
## R2 FALSE  TRUE TRUE
## R3 FALSE  TRUE TRUE
``````

Subsetting with a logical vector in the row or column slot selects the TRUE rows or columns. You can also subset with logical matrices, although it isnt used much.

``````m1[m1 > 4]
``````
``````## [1] 5 6 7 8 9
``````

Here, we just got the vector. This may not be very useful but it may be for assignment.

``````m1[m1 > 4] <- 0
m1
``````
``````##    C1 C2 C3
## R1  1  4  0
## R2  2  0  0
## R3  3  0  0
``````

Operations

Arithmetic operations with matrices are computed component-wise, just like vectors. This can lead to surprising behavior.

``````m1
``````
``````##    C1 C2 C3
## R1  1  4  0
## R2  2  0  0
## R3  3  0  0
``````
``````2 * m1
``````
``````##    C1 C2 C3
## R1  2  8  0
## R2  4  0  0
## R3  6  0  0
``````
``````w <- c(-1, -2, -3)
m1 * w
``````
``````##    C1 C2 C3
## R1 -1 -4  0
## R2 -4  0  0
## R3 -9  0  0
``````

This multipled each column by w, entry by entry.

To perform matrix multiplication, use the %*% operator.

``````m1 %*% w
``````
``````##    [,1]
## R1   -9
## R2   -2
## R3   -3
``````

You may need to transpose a matrix or extract the diagonal. The R functions for those are `t` and `diag`. R also has functions for doing linear algebra, like singular value decomposition.