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 Mathematica HowTo
How to | Update Parts of a Matrix
Mathematica has many matrix operations that support operations such as building, computing and visualizing matrices. It also has a rich language for picking out parts of matrices and assigning new values to them.
Define the following matrix:
 Out[1]//MatrixForm=
Use [[...]] (the short form of Part) on the left-hand side of an assignment to set an element:
 Out[2]=
This shows that the element at position (1, 2) has been updated:
 Out[3]//MatrixForm=
To set an entire row, use one index to specify the row and assign it to the new row:
 Out[4]//MatrixForm=
To set an entire column, select all rows with All and specify the column:
 Out[5]//MatrixForm=

To set a submatrix you can use the short form ;; of Span.
First set up a 5×5 matrix of random integers between 0 and 10:
 Out[6]//MatrixForm=
The top-left 3×4 matrix highlighted here corresponds to rows 1 through 3 and columns 1 through 4:
 Out[7]=
Update the highlighted submatrix by using the short form ;; of Span to specify the relevant span of rows and columns:
 Out[8]//MatrixForm=
Update all elements except the outermost rows and columns (negative indices count from the end):
 Out[9]//MatrixForm=

When you update a large matrix you should try to avoid doing this in a loop. If you can use one of the updating techniques to update all elements in one command, this will typically be much faster:
This is a slow way to update every element in a row:
 Out[11]=
This is much faster:
 Out[12]=
If you cannot avoid updating a matrix in a loop, you need to take care to avoid extra references to the matrix. Otherwise the matrix will be copied and the loop will not run very fast:
This takes care to avoid extra references. The loop runs quite fast:
 Out[14]=
This makes a copy of the matrix at every step, and the loop does not run very fast:
 Out[15]=
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