Appendix A Sage Reference
As we have introduced a number of Sage commands throughout the text, it may be helpful to have the most important ones summarized in a single place. That is the aim of this appendix.
 Accessing Sage

In addition to the Sage cellls included throughout the book, there are a number of ways to access Sage.
There is a freely available Sage cell at
sagecell.sagemath.org
.You can save your Sage work by creating an account at
cocalc.com
and working in a Sage worksheet.There is a page of Sage cells at
gvsu.edu/s/0Ng
. The results obtained from evaluating one cell are available in other cells on that page. However, you will lose any work once the page is reloaded.
 Creating matrices

There are a couple of ways to create matrices. For instance, the matrix
\begin{equation*} \begin{bmatrix} 2 \amp 3 \amp 0 \amp 4 \\ 1 \amp 2 \amp 1 \amp 3 \\ 0 \amp 2 \amp 3 \amp 0 \\ \end{bmatrix} \end{equation*}can be created in either of the two following ways.
matrix(3, 4, [2, 3, 0, 4, 1,2, 1,3, 0, 2, 3, 0])
matrix([ [2, 3, 0, 4], [ 1,2, 1,3], [ 0, 2, 3, 0] ])
Be aware that Sage can treat mathematically equivalent matrices in different ways depending on how they are entered. For instance, the matrix
matrix([ [1, 2], [2, 1] ])
has integer entries while
matrix([ [1.0, 2.0], [2.0, 1.0] ])
has floating point entries.
If you would like the entries to be considered as floating point numbers, you can include
RDF
in the definition of the matrix.matrix(RDF, [ [1, 2], [2, 1] ])
 Special matrices

The \(4\times 4\) identity matrix can be created with
identity_matrix(4)
A diagonal matrix can be created from a list of its diagonal entries. For instance,
diagonal_matrix([3,4,2])
 Reduced row echelon form

The reduced row echelon form of a matrix can be obtained using the
rref()
function. For instance,A = matrix([ [1,2], [2,1] ]) A.rref()
 Vectors

A vector is defined by listing its components.
v = vector([3,1,2])
 Addition

The
+
operator performs vector and matrix addition.v = vector([2,1]) w = vector([3,2]) print(v+w)
A = matrix([[2,3],[1,2]]) B = matrix([[4,1],[3,1]]) print(A+B)
 Multiplication

The
*
operator performs scalar multiplication of vectors and matrices.v = vector([2,1]) print(3*v) A = matrix([[2,1],[3,2]]) print(3*A)
Similarly, the
*
is used for matrixvector and matrixmatrix multiplication.A = matrix([[2,3],[1,2]]) v = vector([2,1]) print(A*v) B = matrix([[4,1],[3,1]]) print(A*B)
 Operations on vectors
The length of a vector
v
is found usingv.norm()
.The dot product of two vectors
v
andw
isv*w
.
 Operations on matrices
The transpose of a matrix
A
is obtained using eitherA.transpose()
orA.T
.The inverse of a matrix
A
is obtained using eitherA.inverse()
orA^1
.The determinant of
A
isA.det()
.A basis for the null space \(\nul(A)\) is found with
A.right_kernel()
.Pull out a column of
A
using, for instance,A.column(0)
, which returns the vector that is the first column ofA
.The command
A.matrix_from_columns([0,1,2])
returns the matrix formed by the first three columns ofA
.
 Eigenvectors and eigenvalues
The eigenvalues of a matrix
A
can be found withA.eigenvalues()
. The number of times that an eigenvalue appears in the list equals its multiplicity.The eigenvectors of a matrix having rational entries can be found with
A.eigenvectors_right()
.
If \(A\) can be diagonalized as \(A=PDP^{1}\text{,}\) then
D, P = A.right_eigenmatrix()
provides the matrices
D
andP
. The characteristic polynomial of
A
isA.charpoly('x')
and its factored formA.fcp('x')
.
 Matrix factorizations

The \(LU\) factorization of a matrix
P, L, U = A.LU()
gives matrices so that \(PA = LU\text{.}\)

A singular value decomposition is obtained with
U, Sigma, V = A.SVD()
It's important to note that the matrix must be defined using
RDF
. For instance,A = matrix(RDF, 3,2,[1,0,1,1,1,1])
. The \(QR\) factorization of
A
isA.QR()
provided thatA
is defined usingRDF
.
