Data Structures, Algorithms, & Applications in Java
Chapter 8, Exercise 29

Since the class UpperTriangularMatrix of Exercise 27 stores an upper-triangular matrix by columns and the class LowerTriangularMatrix of the text stores the matrix by rows, the transpose operation can be done by just copying the nonzero elements as below: 
public class ExtendedLowerTriangularMatrix extends LowerTriangularMatrix
{
   // constructor
   public ExtendedLowerTriangularMatrix(int theRows, Object theZero)
      {super(theRows, theZero);}

   /** return the transpose of this */
   public UpperTriangularMatrix transpose()
   {
      // create result matrix
      UpperTriangularMatrix u = new UpperTriangularMatrix(rows, zero);

      // form the transpose
      for (int i = 0; i < rows * (rows + 1) / 2; i++)
         u.element[i] = element[i];
   
      return u;
   }
}


The time complexity of transpose is Theta(rows2) as rows(rows+1)/2 elements are copied.
The test program and output are in the files ExtendedLowerTriangularMatrix.*.