How to | Create a Matrix

Matrices are represented in the Wolfram Language with lists. They can be entered directly with the { } notation, constructed from a formula, or imported from a data file. The Wolfram Language also has commands for creating diagonal matrices, constant matrices, and other special matrix types.

A matrix can be entered directly with notation:

In[1]:=
Click for copyable input
Out[1]=

You can show the result in matrix notation with MatrixForm:

In[2]:=
Click for copyable input
Out[2]//MatrixForm=

is another way of entering . It can be convenient to use it when is a formatting function.

This uses Table to create a grid of values in and :

In[3]:=
Click for copyable input
In[4]:=
Click for copyable input
Out[4]//MatrixForm=

Note that matrices in the Wolfram Language are not restricted to contain numbers; they can contain symbolic entries such as formulas:

In[5]:=
Click for copyable input
Out[5]//MatrixForm=
    

When you create a matrix and save it with an assignment, take care not to combine this with formatting using MatrixForm. Use parentheses:

In[1]:=
Click for copyable input
Out[1]//MatrixForm=

You can use in further calculations:

In[2]:=
Click for copyable input
Out[2]//MatrixForm=

Suppose you do not use parentheses:

In[3]:=
Click for copyable input
Out[3]//MatrixForm=

Then will print like a matrix but will not work in calculations like a matrix. For example, the following does not carry out matrix multiplication:

In[4]:=
Click for copyable input
Out[4]=

You can check the value of by using FullForm:

In[5]:=
Click for copyable input
Out[5]//FullForm=

This shows that also includes the formatting wrapper MatrixForm, which stops it from working as a matrix.

    

There are functions to create a variety of special types of matrices.

This creates a 4×5 matrix of real values that fall between and :

In[1]:=
Click for copyable input
Out[1]//MatrixForm=

This creates a matrix that only has nonzero entries on the diagonal:

In[2]:=
Click for copyable input
Out[2]//MatrixForm=

This creates a matrix whose entries are all the same:

In[3]:=
Click for copyable input
Out[3]//MatrixForm=

This creates a 4×4 Hilbert matrix; each entry is of the form :

In[4]:=
Click for copyable input
Out[4]//MatrixForm=
    

Many linear algebra and other functions return matrices.

Here, the QR decomposition of a random 3×3 matrix is calculated:

In[1]:=
Click for copyable input

This prints the Q matrix:

In[2]:=
Click for copyable input
Out[2]//MatrixForm=

When Wolfram Language functions return matrices they often use an optimized storage format called packed arrays.

    

You can apply many common operations in the Wolfram Language to a list, and get back another list with the function mapped onto each element. This also works for matrices, which are lists of lists.

Here is a 2×2 matrix of squares:

In[1]:=
Click for copyable input
Out[1]//MatrixForm=

This applies Sqrt to each element of the matrix:

In[2]:=
Click for copyable input
Out[2]//MatrixForm=

This behavior of Sqrt is called listability, and it makes very readable and efficient code.

If a function that is not listable is used, it does not map onto each element:

In[3]:=
Click for copyable input
Out[3]=

You can make the function listable; now it will map onto each element:

In[4]:=
Click for copyable input
Out[4]//MatrixForm=
    

Another important way to create a matrix is to import a data file. This can be done with tabular formats such as Table (.dat), CSV (.csv), and TSV (.tsv). A matrix can also be read from an Excel spreadsheet (.xls).

Here, ImportString is used to import a CSV formatted string into a matrix. Importing from a file is done with Import:

In[1]:=
Click for copyable input
Out[1]=

The Wolfram Language also supports a number of other formats including scientific and medical data formats such as HarwellBoeing, MAT, HDF, NASACDF, and FITS.

    

The way that you create a matrix can have an important impact on the efficiency of your programs. For the best efficiency, avoid appending to a matrix, avoid unnecessary creation operations, and use listable operations when you can.

This example repeatedly adds a new row to a matrix:

In[1]:=
Click for copyable input
Out[1]=

It is much faster to create the matrix in one computation. Whenever you see a For loop, try to replace it with some other construct, like Table:

In[2]:=
Click for copyable input
Out[2]=

The following example creates a matrix of zeros and then fills it in with a loop. The creation of a zero matrix here is completely unnecessary:

In[3]:=
Click for copyable input
Out[3]=

It is much faster to create data for each row of the matrix once, and then use a listable operation:

In[4]:=
Click for copyable input
Out[4]=

If your matrices are large and have many elements that are the same (for example, zero), then you should consider working with sparse matrices formed with SparseArray.