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# 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.

This is a 2×2 matrix.
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Here is the first row.
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Here is the element m12.
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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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Or a matrix by a matrix.
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Or a vector by a matrix.
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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 ab) 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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This gives a 3×2 matrix with symbolic elements. "Building Lists from Functions" will discuss how you can produce other kinds of elements with Array.
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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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 c m multiply a matrix by a scalar a.b dot product of two matrices Inverse[m] matrix inverse MatrixPower[m,n] nth power of a matrix Det[m] determinant Tr[m] trace Transpose[m] transpose Eigenvalues[m] eigenvalues Eigenvectors[m] eigenvectors

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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This gives its inverse.
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Taking the dot product of the inverse with the original matrix gives the identity matrix.
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Here is a 3x3 matrix.
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Eigenvalues gives the eigenvalues of the 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.