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1.8.3 Vectors and Matrices

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

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.

Functions for vectors.

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]=

Some mathematical operations on matrices.

Here is the matrix of symbolic variables that was defined above.

In[18]:= m

Out[18]=

This gives its determinant.

In[19]:= Det[m]

Out[19]=

Here is the transpose of m.

In[20]:= Transpose[m]

Out[20]=

This gives the inverse of m in symbolic form.

In[21]:= Inverse[m]

Out[21]=

Here is a rational matrix.

In[22]:= h = Table[1/(i+j-1), {i, 3}, {j, 3}]

Out[22]=

This gives its inverse.

In[23]:= Inverse[h]

Out[23]=

Taking the dot product of the inverse with the original matrix gives the identity matrix.

In[24]:= % . h

Out[24]=

Here is a matrix.

In[25]:= r = Table[i+j+1, {i, 3}, {j, 3}]

Out[25]=

Eigenvalues gives the eigenvalues of the matrix.

In[26]:= Eigenvalues[r]

Out[26]=

This gives a numerical approximation to the matrix.

In[27]:= rn = N[r]

Out[27]=

Here are numerical approximations to the eigenvalues.

In[28]:= Eigenvalues[rn]

Out[28]=

Section 3.7 discusses many other matrix operations that are built into Mathematica.