Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
WOLFRAM LANGUAGE TUTORIAL

Vectors and Matrices

Vectors and matrices in the Wolfram Language 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.

This is a 2×2 matrix.
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Here is the first row.
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Here is the element .
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This is a twocomponent vector.
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The objects and 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 the Wolfram Language 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 .
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This displays in standard twodimensional 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" discusses 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] n 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 .
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This gives the inverse of 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 3×3 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 the Wolfram Language.