Schur Complement
Partition M=[[a,b],[c,D]] with scalar a=M11 and 2x2 D. Compute S=D-cb/a, eliminate the two first-column entries in one finite block step and verify detM=a detS.
Free theory articlePartition M=[[a,b],[c,D]] with scalar a=M11 and 2x2 D. Compute S=D-cb/a, eliminate the two first-column entries in one finite block step and verify detM=a detS.
Concept & interpretation
The Schur complement of a block matrix helps in converting block matrices into upper triangular block form, making it easier to solve linear systems and compute determinants.
Experiment
In this module, you can adjust the components of a 3x3 matrix M.
- Click Reset. Use M11=2, M12=1, M13=0, M21=1, M22=3, M23=1, M31=0, M32=1, M33=2. All nine named number controls accept integers between -4 and 6.
- Click the Eliminate first block button to perform the block elimination.
- Observe the transformed matrix, the Schur complement S, and the determinant identity: S=[[2.5,1],[1,2]], det S=4, direct det M=8, and a det S=8.
- Use the Reset button to return to the initial state.