Is lower trianglular form of a matrix just the transpose of upper triangular form.?

Answer 1

No, they are very different. If a matrix can be decomposed into an upper and a lower triangular form, then A = LU. This method is known as LU decomposition.

The upper form is done as you would using Gaussian elimination to reduce a matrix to row echelon form. However, you cannot interchange the rows like you normall would.

The lower form is a bit harder to explain. It's basically a "reverse" of what you do to the matrix to reduce it to row echelon form. If you multiply a row by a constant to get that diagonal entry to 1, you would put the constant's reciprocal in the diagonal entry of the lower form. If you add a multiple of a row to get a 0 for a non-diagonal term, you put the negative of that multiplier in the position where it became 0.

I know it's a bit confusing without an example. But I'm sure if you looked up LU decomposition, you'd get sites with more info.