The Mathematica Book Online: A Practical Introduction to Mathematica

The Mathematica Book Online

A Practical Introduction to Mathematica

Lists

1.8.3 Vectors and Matrices

Vectors and matrices in Mathematica are simply represented by lists and by lists of lists, respectively.

{a, b, c} vector
{{a, b}, {c, d}} matrix

The representation of vectors and matrices by lists.

This is a

matrix.

In[1]:= m = {{a, b}, {c, d}}

Out[1]=

Here is the first row.

In[2]:= m[[1]]

Out[2]=

Here is the element

In[3]:= m[[1,2]]

Out[3]=

This is a two-component vector.

In[4]:= v = {x, y}

Out[4]=

The objects p and q are treated as scalars.

In[5]:= p v + q

Out[5]=

Vectors are added component by component.

In[6]:= v + {xp, yp} + {xpp, ypp}

Out[6]=

This takes the dot ("scalar") product of two vectors.

In[7]:= {x, y} . {xp, yp}

Out[7]=

You can also multiply a matrix by a vector.

In[8]:= m . v

Out[8]=

Or a matrix by a matrix.

In[9]:= m . m

Out[9]=

Or a vector by a matrix.

In[10]:= v . m

Out[10]=

This combination makes a scalar.

In[11]:= v . m . v

Out[11]=

Because of the way Mathematica uses lists to represent vectors and matrices, you never have to distinguish between "row" and "column" vectors.

Table[f, {i, n}] build a length- vector by evaluating f with i = 1, 2, ... , n
Array[a, n] build a length- vector of the form {a[1], a[2], ... }
Range[n] create the list {1, 2, 3, ... , n}
Range[, ] create the list {, +1, ... , }
Range[, , dn] create the list {, +dn, ... , }
list[[i]] or Part[list, i] give the i element in the vector list
Length[list] give the number of elements in list
ColumnForm[list] display the elements of list in a column
c v multiply by a scalar
a . b vector dot product
Cross[a, b] vector cross product (also input as a b)
Norm[v] norm of a vector

Functions for vectors.

Table[f, {i, m}, {j, n}] build an matrix by evaluating f with i ranging from 1 to m and j ranging from 1 to n
Array[a, {m, n}] build an matrix with element a[i, j]
IdentityMatrix[n] generate an identity matrix
DiagonalMatrix[list] generate a square matrix with the elements in list on the diagonal
list[[i]] or Part[list, i] give the i row in the matrix list
list[[All, j]] or Part[list, All, j]
give the j column in the matrix list
list[[i, j]] or Part[list, i, j] give the element in the matrix list
Dimensions[list] give the dimensions of a matrix represented by list
MatrixForm[list] display list in matrix form

Functions for matrices.

This builds a

matrix

with elements

In[12]:= s = Table[i+j, {i, 3}, {j, 3}]

Out[12]=

This displays s in standard two-dimensional matrix format.

In[13]:= MatrixForm[s]

Out[13]//MatrixForm=

This gives a vector with symbolic elements. You can use this in deriving general formulas that are valid with any choice of vector components.

In[14]:= Array[a, 4]

Out[14]=

This gives a

matrix with symbolic elements. Section 2.2.6 will discuss how you can produce other kinds of elements with Array.

In[15]:= Array[p, {3, 2}]

Out[15]=

Here are the dimensions of the matrix on the previous line.

In[16]:= Dimensions[%]

Out[16]=

This generates a

diagonal matrix.

In[17]:= DiagonalMatrix[{a, b, c}]

Out[17]=

c m multiply by a scalar
a . b matrix product
Inverse[m] matrix inverse
MatrixPower[m, n] n power of a matrix
Det[m] determinant
Tr[m] trace
Transpose[m] transpose
Eigenvalues[m] eigenvalues
Eigenvectors[m] eigenvectors