10.1: Matrices

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A matrix is a two-dimensional version of a vector. Like a vector, it contains elements that are identified by indices. The difference is that the elements are arranged in rows and columns, so it takes two indices to identify an element.

Creating a Matrix

A common way to create a matrix is the zeros function, which returns a matrix with the given size filled with zeros. This example creates a matrix with two rows and three columns.

>> M = zeros(2, 3)

M =  0     0     0
     0     0     0

If you don’t know the size of a matrix, you can display it by using whos:

>> whos M
  Name      Size            Bytes  Class     Attributes
  M         2x3                48  double

or the size function, which returns a vector:

>> V = size(M)

V = 2    3

The first element is the number of rows; the second is the number of columns.

To read an element of a matrix, you specify the row and column :

>> M(1,2)

ans = 0

>> M(2,3)

ans = 0

When you’re working with matrices, it takes some effort to remember which index comes first, row or column. I find it useful to repeat “row, column” to myself, like a mantra. You might also find it helpful to remember “down, across” or the abbreviation RC as in “radio control” or RC Cola.

Another way to create a matrix is to enclose the elements in brackets, with semicolons between rows:

>> D = [1,2,3 ; 4,5,6]

D =  1     2     3
     4     5     6

>> size(D)

ans = 2     3

Row and Column Vectors

Although it’s useful to think in terms of numbers, vectors, and matrices, from MATLAB’s point of view everything is a matrix. A number is just a matrix that happens to have one row and one column:

>> x = 5;
>> size(x)

ans = 1     1

And a vector is a matrix with only one row:

>> R = 1:5;
>> size(R)

ans = 1     5

Well, some vectors have only one row, anyway. Actually, there are two kinds of vectors. The ones we’ve seen so far are called row vectors, because the elements are arranged in a row; the other kind are column vectors, where the elements are in a single column.

One way to create a column vector is to create a matrix with only one element per row:

>> C = [1;2;3]

C =

     1
     2
     3

>> size(C)

ans = 3     1

The difference between row and column vectors is important in linear algebra, but for most basic vector operations, it doesn’t matter. For example, when you index the elements of a vector, you don’t have to know what kind it is:

>> R(2)

ans = 2

>> C(2)

ans = 2

The Transpose Operator

The transpose operator, which looks remarkably like an apostrophe, computes the transpose of a matrix, which is a new matrix that has all of the elements of the original, but with each row transformed into a column (or you can think of it the other way around).

In this example D has two rows:

>> D = [1,2,3 ; 4,5,6]

D =  1     2     3
     4     5     6

so its transpose has two columns:

>> Dt = D'

Dt = 1     4
     2     5
     3     6

Exercise 10.1

What effect does the transpose operator have on row vectors, column vectors, and numbers?