Question

2. Suppose a matrix has four blocks A, B,C, D, where A is invertible. Block elimination can be done on this block matrix. For example, consider what happens when CA-1 times the first block row [A B] is subtracted from the second block row [C D 0 D - CA B Note: S = D-CA-B is called the Schur complement of A. (a) (3 points) Find X and Y such that AB (b) (3 points) Find S for A B

Answer 1

[A C B D] = [I X O I] [A O O S] [I O Y I] Implies [A C B D] = [I X O I] [AI + OO OI + SO AY + OI OY + SI] Implies [A C B D] = [I X O I] [A O AY S] Implies [A C B D] = [IA + OO XA + IO IAY + OS XAY + IS] implies [A C B D] = [A XA AY XAY + S] therefore By equating corresponding blocks, we get B = AY C = XA Implies AY = B rightarrow (1) XA = C rightarrow (2) Since A is invertible (Given) therefore A^-1 exist Implies AA^-1 = A^-1 A = I pre multiply equation (1) by A^-1, we get A^-1 middot AY = A^-1 B IY = A^-1 B (since A^-1 A = I) Y = A^-1 B Now, past multiply equation (2) A^-1, we get XAA^-1 = CA^-1 XI = CA^-1 (since AA^-1 = I) X = CA^-1 \r\n [A C B I] = [2 4 4 3 1 0