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(TL;DR: matrices are rectangular arrays of numbers.)

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I wanted to avoid a picture related to the film franchise!

Me

Last

time,

we learnt about **dot products**. We will now finally start talking

aobut **matrices**.

# Today’s topic: Matrices – a primer

We know what row vectors and column vectors are from our previous posts.

What would it look like if we were to stack a bunch of row vectors on

top of each other? Similarly, what would it look like if we were to put a

bunch of column vectors next to each other? Let’s experiment!

Let’s define some row vectors:

Let’s stack them on top of each other:

This rectangular array of numbers is an example of a **matrix**!

Let’s repeat the exercise with the same vectors represented as column

vectors:

Putting these column vectors next to each other gives us this:

Looks like another matrix to me!

In fact, our row vectors and column vectors we saw in previous posts

live double lives. A row vector goes by another name: **“row matrix”**.

A column vector goes by another name, too: **“column matrix”**.

Sneaky!

# A little bit of notation

As before, we will be following the notation used in *Goodfellow, Ian,
et al.*. We will

depict matrices using uppercase, bold font:

# Describing the size of a matrix

From the above, we can guess that matrices can be described in terms of

**rows and columns**. How can we describe the **number of rows and the
number of columns** we have in our matrices?

Let’s go back to our previous examples. They both happen to have *three
rows* and

*three columns*. So we have ourselves two

*3 x 3 (“three by*

three”) matrices.

three”) matrices

**The number of rows and the number of columns**

describe the “size” of our matrix.

describe the “size” of our matrix.

Let’s generalise this. By convention, we have rows and columns

in our matrix. Let’s use our matrix as an example. We

haven’t defined what elements it contains. In fact, all we have said is

that is a matrix. Whatever the contents of

, it will will have rows and columns. So

will be of size .

# Keep it real

Let’s take this a step further. From previous posts, we are working with

**real numbers** so let’s assume that our matrix of

size will contain real numbers. That is, it is a **real
matrix**. How can we depict this using some compact notation?

Remember when we touched on our vectors belonging to some abstract

**set** of all real-valued vectors of **similar dimensions** when describing the dimensions of a

vector?

We can say something similar for our real-valued matrix of size

. We can describe this set of all real-valued matrices like this:

So to show that our matrix belongs to the set of all

matrices that are made up of real numbers, we simply say:

Going back to our examples above:

- Our real-valued row vectors (a.k.a.
**row matrices**) are members of

the set . That is, they are members of the

set of all possible matrices made up of one row and columns of

real numbers. - Our real-valued column vectors (a.k.a.
**column matrices**) are

members of the set . That is, they are

members of the set of all possible matrices made up of rows and one column of real numbers. - Our real-valued square matrices are members of the set

where . That is, they are members

of the set of all possible real matrices that have the same number

of rows as they have columns.

Not too bad, right?

# How can we create matrices in R?

## Stack those vectors

One way is to simply `rbind`

and `cbind`

vectors. Let’s define 3 vectors

and `rbind`

them (i.e. stack them row-wise):

```
a <- c(1, 2, 3)
b <- c(4, 5, 6)
c <- c(7, 8, 9)
A <- rbind(a, b, c)
print(A)
```

```
## [,1] [,2] [,3]
## a 1 2 3
## b 4 5 6
## c 7 8 9
```

We have ourselves a matrix!

```
print(class(A))
```

```
## [1] "matrix"
```

Let’s now stack them column-wise by using `cbind`

on our vectors:

```
A <- cbind(a, b, c)
print(A)
```

```
## a b c
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9
```

It’s a matrix!

```
print(class(A))
```

```
## [1] "matrix"
```

## Use the `matrix`

function

The “usual” way to create a matrix would be to pass some vector or list

into the `matrix`

function and specify its dimensions. For example, if we

want to use the values passed into the `matrix`

function to create a 3×3 matrix by using the values in a **column-wise** manner, we can do this:

```
matrix(1:9, nrow=3, ncol=3)
```

```
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9
```

If instead we want to use the numbers in a **row-wise** manner, then we

can use the `byrow`

argument and set it to `TRUE`

:

```
matrix(1:9, nrow=3, ncol=3, byrow=TRUE)
```

```
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9
```

The data types of the elements in the vector or list you pass into the

`matrix`

function will be coerced to a single data type following the

usual coercion rules. For example, if we use a vector containing a

`numeric`

type and a `integer`

type, let’s see what happens:

```
our_vec <- c(1.0, 1L)
A <- matrix(our_vec, nrow=2, ncol=1)
print(A)
```

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

From previous posts, we know that the values in our vector have already

been coerced by `c`

before the `matrix`

function sees them! So,

naturally, we have a matrix of `numeric`

types:

```
sapply(A, class)
```

```
## [1] "numeric" "numeric"
```

We can even have a character matrix. Not sure why we’d want to create

one…but we can!

```
x <- c("hello", "goodbye")
matrix(x)
```

```
## [,1]
## [1,] "hello"
## [2,] "goodbye"
```

## Elements are recycled

Recycling occurs like in vector addition posts. For example, if we try to

create a matrix with four elements (2 rows times 2 columns) from a

vector with only two elements, this is what we get:

```
matrix(c(1, 2), nrow=2, ncol=2)
```

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

If the number of elements in our vector is not a multiple or submultiple

of the number of rows or columns, we get warned of this fact:

```
matrix(c(1, 2, 3), nrow=2, ncol=1)
```

```
## Warning in matrix(c(1, 2, 3), nrow = 2, ncol = 1): data length [3] is not a
## sub-multiple or multiple of the number of rows [2]
## [,1]
## [1,] 1
## [2,] 2
```

Done!

# Conclusion

Next time, we’ll talk about matrix addition and introduce some notation we can use to access the individual elements contained within our matrices!

Justin

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