Mathematica HowTo
How to | Create a Matrix
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:
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You can show the result in matrix notation with MatrixForm:
 Out[2]//MatrixForm=
expr//fun is another way of entering fun[expr]. It can be convenient to use it when fun is a formatting function.
This uses Table to create a grid of values in and :
 Out[4]//MatrixForm=
Note that matrices in Mathematica are not restricted to contain numbers; they can contain symbolic entries such as formulas:
 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:
 Out[6]//MatrixForm=
You can use mat in further calculations:
 Out[7]//MatrixForm=
Suppose you do not use parentheses:
 Out[8]//MatrixForm=
Then mat will print like a matrix but will not work in calculations like a matrix. For example, the following does not carry out matrix multiplication:
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You can check the value of mat by using FullForm:
 Out[10]//FullForm=
This shows that mat 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[11]//MatrixForm=
This creates a matrix that only has nonzero entries on the diagonal:
 Out[12]//MatrixForm=
This creates a matrix whose entries are all the same:
 Out[13]//MatrixForm=
This creates a 4×4 Hilbert matrix; each entry is of the form 1/(i + j - 1):
 Out[14]//MatrixForm=

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[16]//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[17]//MatrixForm=
This applies Sqrt to each element of the matrix:
 Out[18]//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:
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You can make the function listable; now it will map onto each element:
 Out[20]//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:
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Mathematica also supports a number of other formats including scientific and medical data formats such as HarwellBoeing, MAT, HDF, CDF, 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:
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It is much faster to create the matrix in one computation. Whenever you see a For loop you want to see how to replace it with some other construct like Table:
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The following example creates a n matrix of zeros and then fills it in with a loop. The creation of a zero matrix here is completely unnecessary:
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It is much faster to create data for each row of the matrix once, and then use a listable operation:
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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.