## Creating Matrices in Mathematica (2015)

### 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]

### 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

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.

References