The Mathematica Book Online: Advanced Mathematics in Mathematica

The Mathematica Book Online

Advanced Mathematics in Mathematica

Linear Algebra

3.7.1 Constructing Matrices

Table[f, {i, m}, {j, n}] build an matrix where f is a function of i and j that gives the value of the entry
Array[f, {m, n}] build an matrix whose entry is f[i, j]
DiagonalMatrix[list] generate a diagonal matrix with the elements of list on the diagonal
IdentityMatrix[n] generate an identity matrix
Normal[SparseArray[{{, }->, {, }->, ... }, {m, n}]]
make a matrix with non-zero values at positions {, }

Functions for constructing matrices.

This generates a

matrix whose

entry is a[i, j].

In[1]:= Table[a[i, j], {i, 2}, {j, 2}]

Out[1]=

Here is another way to produce the same matrix.

In[2]:= Array[a, {2, 2}]

Out[2]=

DiagonalMatrix makes a matrix with zeros everywhere except on the leading diagonal.

In[3]:= DiagonalMatrix[{a, b, c}]

Out[3]=

IdentityMatrix[n] produces an

identity matrix.

In[4]:= IdentityMatrix[3]

Out[4]=

This makes a

matrix with two non-zero values filled in.

In[5]:= Normal[SparseArray[{{2, 3}->a, {3, 2}->b}, {3, 4}]]

Out[5]=

MatrixForm prints the matrix in a two-dimensional form.

In[6]:= MatrixForm[%]

Out[6]//MatrixForm=

Table[0, {m}, {n}] a zero matrix
Table[Random[ ], {m}, {n}] a matrix with random numerical entries
Table[If[i j, 1, 0], {i, m}, {j, n}]
a lower-triangular matrix

Constructing special types of matrices with Table.

Table evaluates Random[ ] separately for each element, to give a different pseudorandom number in each case.

In[7]:= Table[Random[ ], {2}, {2}]

Out[7]=

SparseArray[{}, {n, n}] a zero matrix
SparseArray[{i_, i_} -> 1, {n, n}]
an identity matrix
SparseArray[{i_, j_}/;ij -> 1, {n, n}]
a lower-triangular matrix

Constructing special types of matrices with SparseArray.

This sets up a general lower-triangular matrix.

In[8]:= SparseArray[{i_, j_}/;ij -> f[i, j], {3, 3}] // MatrixForm

Out[8]//MatrixForm=