Matrices are represented in
Mathematica with lists. They can be entered directly with the { } notation, constructed from a formula, or imported from a data file.
Mathematica also has commands for creating diagonal matrices, constant matrices, and other special matrix types.
A matrix can be entered directly with
notation:
| Out[125]= |  |
You can show the result in matrix notation with
MatrixForm:
Out[126]//MatrixForm= |
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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
:
Out[129]//MatrixForm= |
| |  |
Note that matrices in
Mathematica are not restricted to contain numbers; they can contain symbolic entries such as formulas:
Out[131]//MatrixForm= |
| |  |
When you create a matrix and save it with an assignment, take care not to combine this with formatting using
MatrixForm. Use parentheses:
Out[132]//MatrixForm= |
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You can use
in further calculations:
Out[133]//MatrixForm= |
| |  |
Suppose you do not use parentheses:
Out[134]//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:
| Out[135]= |  |
You can check the value of
by using
FullForm:
Out[11]//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 -10 and 10:
Out[136]//MatrixForm= |
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This creates a matrix that only has nonzero entries on the diagonal:
Out[137]//MatrixForm= |
| |  |
This creates a matrix whose entries are all the same:
Out[138]//MatrixForm= |
| |  |
This creates a 4×4 Hilbert matrix; each entry is of the form
:
Out[139]//MatrixForm= |
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Many linear algebra and other functions return matrices.
Here, the QR decomposition of a random 3×3 matrix is calculated:
This prints the Q matrix:
Out[142]//MatrixForm= |
| |  |
When
Mathematica functions return matrices they often use an optimized storage format called packed arrays.
You can apply many common operations in
Mathematica 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:
Out[144]//MatrixForm= |
| |  |
This applies
Sqrt to each element of the matrix:
Out[145]//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:
| Out[146]= |  |
You can make the function listable; now it will map onto each element:
Out[148]//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:
| Out[149]= |  |
Mathematica 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:
| Out[151]= |  |
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:
| Out[152]= |  |
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:
| Out[153]= |  |
It is much faster to create data for each row of the matrix once, and then use a listable operation:
| Out[154]= |  |
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.
