Chapter 34: Schur Complement¶

Prerequisites: Matrix Algebra (Ch02) · Positive Definite Matrices (Ch16) · Determinants (Ch03)

Chapter Outline: Motivation for Block Elimination → Definition of the Schur Complement → Determinant Decomposition Formula → Inversion Formula for Block Matrices (Banachiewicz Formula) → Criteria for Positive Definiteness → Inertia Additivity (Haynsworth Formula) → Applications in Matrix Equations → Statistical Significance (Partial Correlation and Conditional Variance)

Extension: The Schur complement is the "scalpel" of block matrix computation; it is not only the core of divide-and-conquer solvers for large linear systems but also the mathematical key to understanding Gaussian processes and Kernel methods (Ch29) in modern probability theory.

In handling large-scale systems, we frequently partition matrices into sub-blocks. The Schur Complement is the critical intermediate structure obtained through local elimination. It provides not only explicit expressions for the determinant and inverse of a block matrix but also deeply reveals the correlations between different components of the matrix.

34.1 Definition and Determinant Formula¶

Definition 34.1 (Schur Complement)

Let $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ be a partitioned matrix. If $A$ is invertible, the Schur Complement of $A$ in $M$ is defined as: $$S = D - C A^{-1} B$$

Theorem 34.1 (Determinant Decomposition)

The determinant of the block matrix $M$ can be factored as: $$\det(M) = \det(A) \det(D - C A^{-1} B)$$ This implies that the "volume" of the entire system equals the volume of the top-left subsystem multiplied by the volume of its Schur complement.

34.2 Inversion Formulas for Block Matrices¶

Technique: Banachiewicz Formula

If both $A$ and the Schur complement $S$ are invertible, the inverse of $M$ is: $$M^{-1} = \begin{pmatrix} A^{-1} + A^{-1} B S^{-1} C A^{-1} & -A^{-1} B S^{-1} \\ -S^{-1} C A^{-1} & S^{-1} \end{pmatrix}$$ This is the numerical cornerstone for solving large structured systems.

34.3 Positive Definiteness and Inertia¶

Theorem 34.2 (Positive Definiteness Criterion)

Let $M = \begin{pmatrix} A & B \\ B^T & C \end{pmatrix}$ be symmetric. Then: $$M \succ 0 \iff A \succ 0 \text{ and } C - B^T A^{-1} B \succ 0$$

Theorem 34.3 (Haynsworth Inertia Formula)

The inertia (number of positive, negative, and zero eigenvalues) of $M$ is the sum of the inertia of $A$ and the inertia of its Schur complement $S$: $$\operatorname{In}(M) = \operatorname{In}(A) + \operatorname{In}(D - C A^{-1} B)$$

Exercises¶

Solution

Take $A=(1)$. $S = 5 - 2(1)^{-1}2 = 1$.

Solution

$\det(M) = \det(A)\det(S) = 1 \cdot 1 = 1$.

Solution

$S = D - C A^{-1} 0 = D$. When blocks are decoupled, the Schur complement is simply the original diagonal block.

Solution

$S^T = (D - C A^{-1} B)^T = D^T - B^T (A^{-1})^T C^T = D - C A^{-1} B = S$ (for symmetric $M$, $C = B^T$ and $D^T = D$).

Solution

Using block elimination: $\begin{pmatrix} I & 0 \\ -CA^{-1} & I \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} I & -A^{-1}B \\ 0 & I \end{pmatrix} = \begin{pmatrix} A & 0 \\ 0 & S \end{pmatrix}$. Since the matrices on the left and right are non-singular, the rank is preserved.

Solution

$S$ represents the conditional covariance of the set of variables $D$ given the set of variables $A$.

Solution

$S_D = A - B D^{-1} C$ (assuming $D$ is invertible).

Solution

No. Singular values do not possess the simple block-additivity properties that eigenvalues or inertia indices do.

Solution

$\begin{pmatrix} A^{-1} & 0 \\ -D^{-1} C A^{-1} & D^{-1} \end{pmatrix}$.

Solution

Chapter Summary¶

The Schur complement is the core syntax of block algebra:

****: It demonstrates how to compress the complexity of a high-dimensional system into a low-dimensional remainder using local invertibility—the mathematical prerequisite for distributed algorithms.

****: The Schur complement formula establishes quantitative links between the global stability of a block matrix and the local stability of its subsystems and interaction terms.

****: In probability theory, the Schur complement reveals the algebraic essence of the conditioning process in Gaussian distributions, serving as the ultimate tool for handling "remaining correlation" between variables.