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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 particular 
rational matrix known as a "Hilbert 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 other matrix operations that are built into Mathematica.
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