8: Appendix A Sage Reference

We have introduced a number of Sage commands throughout the text, and the most important ones are summarized here in a single place.

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

\[\left[\begin{array}{cccc}
-2 & 3 & 0 & 4 \\
1 & -2 & 1 & -3 \\
0 & 2 & 3 & 0
\end{array}\right]\]

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  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

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 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 \(A=P D P^{-1}\), 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 .

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