Schur Complement

Concept Overview

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.

Worked Example

Consider the default matrix M. We partition M with a = M11 = 2, b = [1, 0], c = [[1], [0]], and D = [[3, 1], [1, 2]]. The Schur complement S = D - (c * b) / a can be calculated by performing standard arithmetic on these values. This is an important step in block elimination.

M = [[2, 1, 0], [1, 3, 1], [0, 1, 2]] S = [[3, 1], [1, 2]] - ([[1], [0]] * [1, 0]) / 2 = [[2.5, 1], [1, 2]]

The determinant of S is (2.5 * 2) - (1 * 1) = 5 - 1 = 4. The determinant of M is a * det(S) = 2 * 4 = 8. An independent cofactor expansion gives 2(3·2−1·1) − 1(1·2−1·0) + 0 = 10−2 = 8, matching a det(S).

Experiment

In this module, you can adjust the components of a 3x3 matrix M.

1. 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.
2. Click the Eliminate first block button to perform the block elimination.
3. 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.
4. Use the Reset button to return to the initial state.

Limitations

This visualization relies on standard JavaScript floating-point numbers. Rounding errors may occur for non-integer intermediate results. Furthermore, the selected block formula is undefined if M11 is zero, so the elimination will not proceed in that case, even if the entire matrix is invertible through alternative pivoting. It also only demonstrates a fixed 3x3 case, limiting generalization to larger matrices.

References

The following reference describes the mathematical foundation of the Schur complement, showing its properties and applications in linear algebra.

• What Is the Schur Complement of a Matrix? (Nicholas Higham)