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) |
| {{a,b},{c,d}} | matrix  |
The representation of vectors and matrices by lists.
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This is a two-component vector.
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The objects p and q are treated as scalars.
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Vectors are added component by component.
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This gives the dot (scalar) product of two vectors.
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You can also multiply a matrix by a vector.
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This combination makes a scalar.
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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 i=1, 2, ..., n |
| Array[a,n] | build a length-n vector of the form {a[1], a[2], ...} |
| Range[n] | create the list {1, 2, 3, ..., n} |
| Range[n1,n2] | create the list {n1, n1+1, ..., n2} |
| Range[n1,n2,dn] | create the list {n1, n1+dn, ..., n2} |
| list[[i]] or Part[list,i] | give the ith 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 a b) |
| 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 i, jth element a[i, j] |
| IdentityMatrix[n] | generate an m×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 ith row in the matrix list |
| list[[All,j]] or Part[list,All,j] |
| give the jth column in the matrix list |
| list[[i,j]] or Part[list,i,j] | give the i, jth 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 s with elements sij=i+j.
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This displays s in standard two-dimensional matrix format.
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This gives a vector with symbolic elements. You can use this in deriving general formulas that are valid with any choice of vector components.
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Here are the dimensions of the matrix on the previous line.
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This generates a 3×3 diagonal matrix.
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Some mathematical operations on matrices.
Here is the 2×2 matrix of symbolic variables that was defined.
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This gives its determinant.
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Here is the transpose of m.
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This gives the inverse of m in symbolic form.
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Here is a 3×3 rational matrix.
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Taking the dot product of the inverse with the original matrix gives the identity matrix.
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This gives a numerical approximation to the matrix.
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Here are numerical approximations to the eigenvalues.
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"Linear Algebra in Mathematica" discusses many other matrix operations that are built into
Mathematica.
