Matrices

Section 2.2 Matrices

In this section, we will see how to define matrices, and perform basic operations on them. A matrix is an \(m \times n\) rectangular array of numbers:

\begin{equation*} \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} \end{equation*}

where \(m\) is the number of rows and \(n\) is the number of columns. Each entry \(a_{ij}, i=1\dots m, j=1\dots n,\) corresponds to the value at the intersection of the \(i\)-th row and \(j\)-th column.

Just like vectors, Sage does come with built-in support for matrices. There are many ways to define a matrix in Sage using the command matrix, passing for instance each of the rows of the matrix as a list of numbers.

Or as a list of vectors.

Alternatively, we can create a matrix by passing the entries as a list, and the dimensions as arguments to the matrix command. Here is an example of a \(2 \times 3\) matrix

We do not need to specify both dimensions of a matrix. Sage can infer the missing dimension from the number of entries in the list. Here is an example of a \(3 \times 4\) matrix containing the first \(100\) integers, and where only the number of rows is specified.

The entries of the matrix can also be defined programmatically using a list comprehension.

Sage also allows to construct a larger matrix by combining smaller submatrices using the block_matrix() function (or matrix.block()). Submatrices can be arranged in rows and columns to form a single, larger matrix.

In this example, the resulting matrix \(M\) is a 4x4 matrix built from the submatrices \(A\text{,}\) \(-A\text{,}\) \(A^T\) (transpose of \(A\)), and \(100*A\text{.}\) Matrices of different dimensions can be used with the command block_matrix as long as the dimensions result in a uniform rectangular or square matrix.

The individual entries of a matrix can be accessed using the row and column indices. Since indices in Sage start at 0, to retrieve the entry a_{i,j} in a matrix M, we type M[i-1, j-1]. Here is, for instance, how we can retrieve the entry at the second row and third column of the matrix \(M\text{.}\)

Sage provides dedicated methods to retrieve all rows of the matrix, or all its columns. It also support the retrieval of the diagonal elements of the matrix, or a specific entry at a particular row and column.

The rows method returns all rows of the matrix (as a list of tuples).

In the same way, the columns method returns all the columns of the matrix (as a list of tuples).

The M.row(i-1) returns the \(i\)-th row of the matrix.

The M.column(j-1) returns the \(j\)-th column of the matrix.

The M.diagonal() returns the main diagonal of the matrix.

Similarly, Sage offers more methods to help us iterate over the rows and/or the columns of a matrix. For instance, the method nrows() returns the number of rows of a matrix.

Sage method ncols() returns the number of rows of the matrix.

Alternatively, the method dimensions returns the dimensions of the matrix as a pair (nrows, ncols).

To extract a submatrix from a matrix given the range of rows and columns to include, Sage command matrix_from_rows_and_columns() can be used. It takes two lists: the first for row indices, and the second for column indices as in the following example.

An alternative, and perhaps more straightforward way to achieve the same result is to use Python-like slicing and explicitly listing the range of interest as in the following example.

Note that a matrix with a single list of values creates a vector-like object, but it is NOT a vector. Here is a vector in Sage.

Compare that to this single-row matrix. Observe the square brackets in matrices in lieu of the parenthesis.

Type assertion is a useful tool to check if two objects are identical and are of the same type. For instance, comparing the types of \(v\) and \(m\) yields False because they are NOT of the same type.

Just like Vectors, Matrices in Sage can also be created in different Fields, inferred from the datatype passed to the matrix command, or explicitly when instantiating the matrix.

The method base_ring returns the base ring of the matrix.

Once again, if the field is omitted, Sage will simply infers the field from the datatype of the matrix entries. Sage matrices supports the same fields as vectors (\(ZZ, QQ, RR, CC\)).

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