Let be written in block form

where . Let . Assume that has an decomposition. After steps of Gaussian elimination, we have

.

Prove .

No idea where to start on a non-horrible-looking proof. Has anyone seen this theorem before?

Printable View

- October 7th 2012, 02:54 AMskeptopotamusSchur Complement Problem (Spectral/Num. Lin./Matrix Theory)
Let be written in block form

where . Let . Assume that has an decomposition. After steps of Gaussian elimination, we have

.

Prove .

No idea where to start on a non-horrible-looking proof. Has anyone seen this theorem before? - October 7th 2012, 06:08 PMskeptopotamusRe: Schur Complement Problem (Spectral/Num. Lin./Matrix Theory)
Sorry; I should say that it is pretty obvious, but the "obvious proof" is extremely ugly and inelegant. Does anyone have a nicer one?

- October 8th 2012, 09:42 PMskeptopotamusRe: Schur Complement Problem (Spectral/Num. Lin./Matrix Theory)
Ok, I have solved the problem. Here is the proof, in case anybody was interested.

Let such that has an decomposition. Perform Gaussian elimination on up to the th step. Then , . By , we have that .

We have that , where

, and

Note that is invertible by triangular with non-zero entries on the main diagonal. Thus we have that

By block multiplication, this implies that

.

But we wish to show that ,

hence we need to show that ,

i.e. ,

where is the inverse of

by .

Thus

.

But

by .

Therefore,

by . Thus the result is proven.