Question

3. [2+2pt] Let n > 2. Consider a matrix A E Rnxn for which every leading principal submatrix of order less than n is non-singular. (a) Show that A can be factored in the form A = LDU, where Le Rnxn is unit lower triangular, D e Rnxn is diagonal and U E Rnxn is unit upper triangular. (b) If the factorization A = LU is known, where L is unit lower triangular and U is upper triangular, show how to find the LU-factors of the transpose A". Note that our requirement for an LU-factorization is that L is unit lower triangular, and U is upper triangular.

Answer 1

esi 032 Given that soulion Any matrix can be represented in form of Cou where lower unit triangular u= unit upper triangular 0- Diagonal element matrix Suppose 3x3 Matrix (A 3x3= LOU Where LE o D₂ ၃) ၀ - elgi D3 Uie 0,3 Us 423 [ 1 0 Let A= [ 3 Then by form of Matrix Operation on it we will find it in U & L

02 [: :) O 3 O 1 when lower triangular L- a unit dzi 1 13, 1320 0, 03 U22 U₂3 033 U12 a1 013 Uz 023 1 оо 422 aş :) (1 a99 023 . O اپل 433 031 032 033 Luz, 32 au, Urz- 012' 013-9,3 : اول all U2 2 = - aaa 121 U2 - U11 U23 : 923 - Do, 0,3) 131 - 930 d32= 232-f3, 0,2 U 22 آی آی آر) : لول 0 O 432 U12 U22 JC U13 U23 U33 This is factor of AT. Thank you,