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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
\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])
	  
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 \(\nul(A)\) 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.