Matrix Operation

Creating Matrices in Mathematica (2015)

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In this article, we will discuss creating matrices in Mathematica.

Matrices in Mathematica

A matrix is an array of numbers arranged in rows and columns. In Mathematica, matrices are expressed as a list of rows, each of which is a list itself. It means a matrix is a list of lists. If a matrix has $n$ rows and $m$ columns then we call it an $n$ by $m$ matrix. The value(s) in the ith row and jth column is called the $i,j$ entry.

In Mathematica, matrices can be entered with the { } notation, constructed from a formula, or imported from a data file. There are also commands for creating diagonal matrices, constant matrices, and other special matrix types.

Creating Matrices in Mathematica

  • Create a matrix using { } notation
    mat={{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}
    but the output will not be in matrix form, to get in matrix form use commands like
    mat//MatrixForm
  • Creating matrix using Table command
    mat1=Table[b{row, column},
    {row, 1, 4, 1}, {column, 1, 2, 1}]
    ];
    MatrixForm[mat1]
  • Creating symbolic matrices such as
    mat2=Table[xi+xj , {i, 1, 4}, {j, 1, 3}]
    mat2//MatrixForm
  • Creating a diagonal matrix with nonzero entries at its diagonal
    DiagonalMatrix[{1, 2, 3, r}]//MatrixForm
  • Creating a matrix with the same entries i.e. a constant matrix
    ConstantArray[3, {2, 4}]//MatrixForm
  • Creating an identity matrix of order $n\times n$
    IdentityMatrix[4]
Matrices and Mathematica

Matrix Operations in Mathematica

In Mathematica, matrix operations can be performed on both numeric and symbolic matrices.

  • To find the determinant of a matrix
    Det[mat]
  • To find the transpose of a matrix
    Transpose[mat]
  • To find the inverse of a matrix for a linear system
    Inverse[mat]
  • To find the Trace of a matrix i.e. sum of diagonal elements in a matrix
    Tr[mat]
  • To find the Eigenvalues of a matrix
    Eigenvalues[mat]
  • To find the Eigenvector of a matrix
    Eigenvector[mat]
  • To find both Eigenvalues and Eigenvectors together
    Eigensystem[mat]

Note that +, *, and ^ operators all automatically work element-wise.

Displaying Matrix and its Elements

  • mat[[1]]         displays the first row of a matrix where mat is a matrix created above
  • mat[[1, 2]]     displays the element from the first row and second column, i.e. m12 element of the matrix
  • mat[[All, 2]]  displays the 2nd column of matrix

Interactive Input (Menu)

  1. Go to Insert > Table/Matrix > New…
  2. Select Matrix (List of lists).
  3. Define the number of rows and columns.
  4. Click OK.
  5. Use the provided interface to enter values in each cell.

Predefined Matrices

Mathematica provides functions to generate specific types of matrices:

  • IdentityMatrix: Creates an identity matrix.
  • DiagonalMatrix: Creates a diagonal matrix from a specified list.
  • HilbertMatrix: Generates a Hilbert matrix.
  • VandermondeMatrix: Creates a Vandermonde matrix.

Importing from Files

  • Use the Import function to read data from various file formats like CSV, TSV, or Excel spreadsheets and convert them into matrices.
Matrices in Mathematica

References

R Frequently Asked Questions