Schur Complements and Block Matrix Definiteness

Why Block Elimination Matters

A matrix written in blocks can often be simplified by eliminating one group of variables while retaining an exact description of what remains. The Schur complement is the matrix left behind by this elimination. It appears naturally in block inverse calculations and, for symmetric matrices, gives a precise test for positive definiteness. The key is to keep track of which block is being eliminated and to check that it is invertible.

Definition and Block Elimination

Let \(A\) be a \(p\times p\) invertible matrix, \(B\) a \(p\times q\) matrix, \(C\) a \(q\times p\) matrix, and \(D\) a \(q\times q\) matrix. Partition a square matrix \(M\) conformably:

The Schur complement of \(A\) in \(M\) is the \(q\times q\) matrix \(D-CA^{-1}B\). It describes the lower-right block after eliminating the terms that couple it to the \(A\) block. In the symmetric case, where \(C=B^{T}\), this complement is \(D-B^{T}A^{-1}B\).

The elimination can be seen directly by multiplying by block triangular matrices. When \(A\) is invertible,

The lower-left block becomes zero because \(-CA^{-1}A+C=0\). The lower-right block is \(D-CA^{-1}B\), exactly the Schur complement. This is the block analogue of eliminating entries below a pivot in ordinary Gaussian elimination.

Block Inverse Formula

Elimination also gives an inverse formula. The formula requires both \(A\) and its Schur complement \(S_A\) to be invertible. Invertibility of \(A\) alone does not guarantee that \(M\) is invertible.

To prove the formula, factor \(M\) using the Schur complement:

Multiplying the factors gives top row \([A\ \ B]\) and bottom row \([C\ \ CA^{-1}B+S_A]=[C\ \ D]\), as required. The second factor can be written as \(\begin{bmatrix}A&0\\0&S_A\end{bmatrix} \begin{bmatrix}I&A^{-1}B\\0&I\end{bmatrix}\). All three factors are invertible under the stated assumptions. Their inverses, in reverse order, are

Multiplying these three matrices gives the displayed block inverse: the upper-left block is \(A^{-1}+A^{-1}BS_A^{-1}CA^{-1}\), the upper-right is \(-A^{-1}BS_A^{-1}\), the lower-left is \(-S_A^{-1}CA^{-1}\), and the lower-right is \(S_A^{-1}\). Thus the product is the inverse of \(M\). \(\square\)

Positive Definiteness Through the Schur Complement

For a symmetric block matrix, the Schur complement has a second use: it separates a quadratic form into a contribution from the pivot block and a contribution from the remaining variables. That decomposition gives an exact criterion for positive definiteness.

For the proof, symmetry of \(A\) and its invertibility imply that \(A^{-1}\) is symmetric. Direct block multiplication gives the congruence factorization

The first and third factors are transposes of one another, and the third factor is invertible. For any vectors \(x\in\mathbb{R}^{p}\) and \(y\in\mathbb{R}^{q}\), this yields

If \(A\) and \(S_A\) are positive definite, the right-hand side is positive for every nonzero \((x,y)\). Indeed, if \(y\ne0\), its second term is positive and its first term is nonnegative; if \(y=0\) and \(x\ne0\), its first term is positive. Thus \(M\) is positive definite.

Conversely, the congruence factorization shows that \(M\) is positive definite exactly when \(\operatorname{diag}(A,S_A)\) is positive definite: an invertible change of variables preserves strict positivity of the quadratic form. The block diagonal matrix is positive definite exactly when each diagonal block is positive definite. To see necessity, test vectors of the forms \((x,0)\) and \((0,y)\); sufficiency follows by adding the two positive quadratic forms for any nonzero \((x,y)\). Therefore \(A\) and \(S_A\) must both be positive definite. \(\square\)

The same criterion can be applied by pivoting on \(D\) instead, when \(D\) is invertible: \(M\) is positive definite if and only if \(D\) and \(A-BD^{-1}B^{T}\) are positive definite. The choice of pivot is a calculation choice; it does not change the matrix being tested.

Common Pitfalls and Practical Use

A frequent error is to call \(D-CA^{-1}B\) “the” complement without saying which block was eliminated. The expression is the Schur complement of \(A\), and it requires \(A\) to be invertible. If the partition or pivot changes, the appropriate formula changes too. Dimensions provide a useful check: \(CA^{-1}B\) must have the same \(q\times q\) dimensions as \(D\).

Another mistake is to infer that a symmetric block matrix is positive definite merely because \(A\) is positive definite. The worked example with \(M=\begin{bmatrix}1&2\\2&1\end{bmatrix}\) disproves that inference. The exact test requires both \(A\) and \(S_A\) to be positive definite. The theorem is stated for strict positive definiteness with an invertible pivot; it should not be transferred unchanged to singular pivots or to a positive-semidefinite claim.

Schur complements matter because they turn block calculations into familiar smaller-matrix calculations. They can reduce an inverse problem to two inverses, identify whether a symmetric block matrix is positive definite, and clarify how coupling between blocks changes the remaining quadratic form. In applications, blocks often represent different groups of variables or parameters; eliminating one group leaves the adjusted contribution of the others in the Schur complement.