## AppendixASage 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.

1. There is a freely available Sage cell at sagecell.sagemath.org.

2. You can save your Sage work by creating an account at cocalc.com and working in a Sage worksheet.

3. 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.

1. matrix(3, 4, [-2, 3, 0, 4,
1,-2, 1,-3,
0, 2, 3, 0])

2. 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])


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 matrix-vector and matrix-matrix 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
1. The length of a vector v is found using v.norm().

2. The dot product of two vectors v and w is v*w.

Operations on matrices
1. The transpose of a matrix A is obtained using either A.transpose() or A.T.

2. The inverse of a matrix A is obtained using either A.inverse() or A^-1.

3. The determinant of A is A.det().

4. A basis for the null space $$\nul(A)$$ is found with A.right_kernel().

5. Pull out a column of A using, for instance, A.column(0), which returns the vector that is the first column of A.

6. The command A.matrix_from_columns([0,1,2]) returns the matrix formed by the first three columns of A.

Eigenvectors and eigenvalues
1. The eigenvalues of a matrix A can be found with A.eigenvalues(). The number of times that an eigenvalue appears in the list equals its multiplicity.

2. The eigenvectors of a matrix having rational entries can be found with A.eigenvectors_right().

3. If $$A$$ can be diagonalized as $$A=PDP^{-1}\text{,}$$ then

D, P = A.right_eigenmatrix()


provides the matrices D and P.

4. The characteristic polynomial of A is A.charpoly('x') and its factored form A.fcp('x').

Matrix factorizations
1. The $$LU$$ factorization of a matrix

P, L, U = A.LU()


gives matrices so that $$PA = LU\text{.}$$

2. 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]).

3. The $$QR$$ factorization of A is A.QR() provided that A is defined using RDF.