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{{a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b,
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3.5 Tables and Matrices
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Mathematica represents tables and matrices as nested lists. Internally, there is no difference in the way they are stored, but they are represented differently using the functions MatrixForm and TableForm. It is often more convenient to use //MatrixForm or //TableForm to the right of the matrix or table name.
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MatrixForm[list] prints double nested lists as a rectangular array enclosed within parentheses. The innermost lists are printed as rows. Single nested lists are printed as columns enclosed within parentheses. TableForm[list] prints list the same way as MatrixForm except the surrounding parentheses are omitted.
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Matrices and tables can be entered directly as nested lists. A matrix or table having m rows and n columns would be a nested list of m sublists, each containing n entries.
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EXAMPLE 42
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list = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}; MatrixForm[list] 1 2 3 4 5 6 7 8 9 10 11 12 or list //MatrixForm
Lists
TableForm[list] or list //TableForm 1 2 3 4 5 6 7 8 9 10 11 12
Matrices and tables can also be conveniently entered by going to Insert Table/Matrix New.
Clicking OK yields an empty grid use the [TAB] key to cycle from entry to entry.
EXAMPLE 43
list = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}};
{{1,
2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}} or list //MatrixForm
MatrixForm[list] 1 2 3 4 5 6 7 8 9 10 11 12
Two special matrix-generating commands are worth remembering because of their frequency in applications. These generate a nested list. Use MatrixForm to get a matrix.
IdentityMatrix[n] produces an n n matrix with 1s on the main diagonal and 0s elsewhere. DiagonalMatrix[list] creates a diagonal matrix whose diagonal entries are the elements of list.
Lists
EXAMPLE 44
IdentityMatrix[3] //MatrixForm 1 0 0 0 1 0 0 0 1 DiagonalMatrix[{1, 2, 3}] //MatrixForm 1 0 0 0 2 0 0 0 3
Once defined, matrices can be combined using the operations of addition, subtraction, scalar, and matrix multiplication. The operation of matrix multiplication is represented by a period (.). Matrices are discussed in greater detail in 12.
EXAMPLE 45
1 2 3 A = 4 5 6 7 8 9
{{1,
The matrix is input using Insert Table/Matrix New. Mathematica outputs the matrix as a nested list of numbers.
2, 3},{4, 5, 6},{7, 8, 9}}
2 1 5 B = 4 7 2 1 3 2
{{2,
1, 5},{4, 7, 2},{1, 3, 2}}
A + B //MatrixForm 3 3 8 8 12 8 8 11 11 A B //MatrixForm 1 1 2 0 2 4 6 5 7 3 A //MatrixForm 3 6 9 12 15 18 21 24 27 A.B //MatrixForm 13 24 15 34 57 42 55 90 69
It is useful to remember that if list is a simple list of numbers, list.list yields the sum of their squares. The result is printed as a single number without braces.
EXAMPLE 46
list = {1, 2, 3, 4, 5}; list.list 55
Lists
Tables are also stored as nested lists, but are represented as tables with TableForm. Although this command allows representation of tables of any dimension, we will discuss only one- and two-dimensional tables in this book.
EXAMPLE 47
list = {{12, 7, 10}, {105, 205, 7}, {3, 30, 300}};
list//TableForm
12 105 3
7 205 30
10 7 300
TableForm[list, options] allows the use of various formatting options in determining the appearance of a table.
From Example 47 we can observe that the numbers in a table are, by default, left justified. This can sometimes make the table confusing to read. Justification can be controlled with the TableAlignments option. TableAlignments Left justifies the columns to the left (default). TableAlignments Right justifies the columns to the right. TableAlignments Center centers the columns.
EXAMPLE 48
list = {{12,7,10},{105,205,7},{3,30,300}}; TableForm[list, TableAlignments Right] 12 7 10 105 205 7 3 30 300
TableForm[list, TableAlignments Center]
12 105 3 7 205 30 10 7 300
Row and column headings can be inserted by using the option TableHeadings within the TableForm command. The default is TableHeadings None. TableHeadings Automatic produces consecutive integer labels for both rows and columns. Each row and column of a table can be labeled separately using strings (characters enclosed within double quotes) or Mathematica expressions. The general form of this option is TableHeadings {rowlist, columnlist} where rowlist is a list of row labels and columnlist is a list of column labels. If you desire to have row labels but not column labels, or column labels but not row labels, simply replace rowlist or columnlist by None.
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