Sage Reference

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 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

The length of a vector v is found using v.norm().
The dot product of two vectors v and w is v*w.

Operations on matrices

The transpose of a matrix A is obtained using either A.transpose() or A.T.
The inverse of a matrix A is obtained using either A.inverse() or A^-1.
The determinant of A is A.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 of A.
The command A.matrix_from_columns([0,1,2]) returns the matrix formed by the first three columns of A.

Eigenvectors and eigenvalues

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.
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 and P.
The characteristic polynomial of A is A.charpoly('x') and its factored form A.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 is A.QR() provided that A is defined using RDF.