Matrix Operations

• Transpose --
  • Algorithm: -- exchange the row and column indices.
• Addition/Subtraction --
  • Precondition: A and B have the same size.
  • Algorithm: -- add/subtract element by element.
  • # operations
• Scalar product --
  • Precondition: c is a real number.
  • Algorithm: -- multiply each element of A by c.
  • # operations
• Dot product (of vectors) --
  • Preconditions:
    • A row vector ( ) times a column vector (v);
    • Two vectors have the same size (n).
  • Algorithm:
  • # operations
  • Norm ( -norm) of a vector -- .
• Matrix product --
  • Precondition: # columns of A equals # rows of B.
  • Algorithm:
  • Alternatively, write A as an array of row vectors

    

    and B as an array of column vectors

    

    then

    

  • # operations :
    • involves dot products, and
    • Each dot product needs O(n) operations.