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.
| Out[1]= |  |
| Out[2]= |  |
Here is the element
.
| Out[3]= |  |
This is a two-component vector.
| Out[4]= |  |
The objects
and
are treated as scalars.
| Out[5]= |  |
Vectors are added component by component.
| Out[6]= |  |
This gives the dot (scalar) product of two vectors.
| Out[7]= |  |
You can also multiply a matrix by a vector.
| Out[8]= |  |
| Out[9]= |  |
| Out[10]= |  |
This combination makes a scalar.
| 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-n vector by evaluating f with  |
| Array[a,n] | build a length-n vector of the form  |
| Range[n] | create the list  |
| Range[n1,n2] | create the list  |
| Range[n1,n2,dn] | create the list  |
| list[[i]] or Part[list,i] | give the i element in the vector list |
| Length[list] | give the number of elements in list |
| c v | multiply a vector by a scalar |
| a.b | dot product of two vectors |
| Cross[a,b] | cross product of two vectors (also input as  ) |
| Norm[v] | Euclidean norm of a vector |
Functions for vectors.
| Table[f,{i,m},{j,n}] | build an m×n matrix by evaluating f with i ranging from 1 to m and j ranging from 1 to n |
| Array[a,{m,n}] | build an m×n matrix with   element  |
| IdentityMatrix[n] | generate an n×n identity matrix |
| DiagonalMatrix[list] | generate a square matrix with the elements in list on the main 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 |
Functions for matrices.
| Column[list] | display the elements of list in a column |
| MatrixForm[list] | display list in matrix form |
Formatting constructs for vectors and matrices.
This builds a 3×3 matrix
with elements
.
| Out[12]= |  |
This displays
in standard two-dimensional matrix format.
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.
| Out[14]= |  |
| Out[15]= |  |
Here are the dimensions of the matrix on the previous line.
| Out[16]= |  |
This generates a 3×3 diagonal matrix.
| Out[17]= |  |
Some mathematical operations on matrices.
Here is the 2×2 matrix of symbolic variables that was defined.
| Out[18]= |  |
This gives its determinant.
| Out[19]= |  |
Here is the transpose of
.
| Out[20]= |  |
This gives the inverse of
in symbolic form.
| Out[21]= |  |
Here is a 3×3 rational matrix.
| Out[22]= |  |
| Out[23]= |  |
Taking the dot product of the inverse with the original matrix gives the identity matrix.
| Out[24]= |  |
| Out[25]= |  |
| Out[26]= |  |
This gives a numerical approximation to the matrix.
| Out[27]= |  |
Here are numerical approximations to the eigenvalues.
| Out[28]= |  |
"Linear Algebra in Mathematica" discusses many other matrix operations that are built into Mathematica.
power of a matrix
