Upper and Lower Triangular Matrices

p {margin: 0; padding: 0;} No Date Upper and Lower Triangular Marrices A square (nrn) matrix C is : - Upper Triangular (UT) FE C 2 0 for , >j 5 (in the diagonal elements are all equal to 1, L VT we have a upper - unit triangular (OUT) - Lower Triangula (CT) it C = o For iej (if the diagonal elements are all equal to 1, 2 0 LT we have a lower unit triangular (LUT) matrix) I 20 - Diagonal (D) iff c =0 for i+j Theoren The product of the two UT (UT) is UT (UUT) The product of the two LT (WT) matrices is LT (LUT) The product of the two D Matrices is D Q = why are we interested in these matrices ? Suppose Ax d, where A is LT (with non-zero diagonel term) Then the solutions are recoursive (forward substitution) Example: X, = d. Azi X, + arr X2 = d2 Az, X, + ase 4 a 55 x 3 = d3 Similarly, support Axid, where A is UT(with non-zero diagonal terms) Then, the solution are recursive ( backward substitution) Example a, X, + as X2 to A13 X3 : d, are X2 + 123 X3 cd2 a3 X3, d3