Math 790-92.01: Matrix Decompositions and Data

This is the webpage for the Spring 2019 manifestation of Dr. Fitzpatrick's section of Math 790-92.01 at Duke University.

SageMath

SageMath is an open source computer algebra system written python.

Useful Commands

Use the syntax A = matrix([(1/2,-3,5),(-4/5,2,11),(6,-4,9)]) to define the matrix \[ A = \left[\begin{array}{rrr} \frac{1}{2} & -3 & 5 \\ -\frac{4}{5} & 2 & 11 \\ 6 & -4 & 9 \end{array}\right] \] Avoid using decimal notation! Input 1/2 instead of .5.

  • The command A.rref() computes the reduced row-echelon form of A.
  • The command A.det() computes the determinant of A, provided that A is square.
  • The command A.rank() computes the rank of A.
  • If A and B are \(m\times n\) matrices, then the command A+B computes the sum of A and B.
  • If c is a scalar, then the command c*A computes the scalar product of c and A.
  • If A is \(\ell\times m\) and B is \(m\times n\), then A*B computes the product of A and B.
  • The command A.transpose() computes the transpose of A.
  • The command A.is_symmetric() tests if A is symmetric.

\(PA=LU\)-Factorizations

The following code computes the \(LU\)-Factorization of a given matrix \(A\).