Schur complement

In linear algebra and the theory of matrices, the Schur complement of a matrix block (i.e., a submatrix within a larger matrix) is defined as follows. Suppose A, B, C, D are respectively p×p, p×q, q×p and q×q matrices, and D is invertible. Let

\( M=\left[\begin{matrix} A & B \\ C & D \end{matrix}\right] \)

so that M is a (p+q)×(p+q) matrix.

Then the Schur complement of the block D of the matrix M is the p×p matrix

\( A-BD^{-1}C.\, \)

It is named after Issai Schur who used it to prove Schur's lemma, although it had been used previously.[1] Emilie Haynsworth was the first to call it the Schur complement.[2]

Background

The Schur complement arises as the result of performing a block Gaussian elimination by multiplying the matrix M from the right with the "block lower triangular" matrix

\( L=\left[\begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right]. \)

Here Ip denotes a p×p identity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is

\( \begin{align} ML &= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]\left[\begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right] = \left[\begin{matrix} A-BD^{-1}C & B \\ 0 & D \end{matrix}\right] \\ &= \left[\begin{matrix} I_p & BD^{-1} \\ 0 & I_q \end{matrix}\right] \left[\begin{matrix} A-BD^{-1}C & 0 \\ 0 & D \end{matrix}\right]. \end{align} \)

That is, we have shown that

\begin{align} \left[\begin{matrix} A & B \\ C & D \end{matrix}\right] &= \left[\begin{matrix} I_p & BD^{-1} \\ 0 & I_q \end{matrix}\right] \left[\begin{matrix} A-BD^{-1}C & 0 \\ 0 & D \end{matrix}\right] \left[ \begin{matrix} I_p & 0 \\ D^{-1}C & I_q \end{matrix}\right], \end{align} \)

and inverse of M thus may be expressed involving D−1 and the inverse of Schur's complement (if it exists) only as

\( \begin{align} & {} \quad \left[ \begin{matrix} A & B \\ C & D \end{matrix}\right]^{-1} = \left[ \begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right] \left[ \begin{matrix} (A-BD^{-1}C)^{-1} & 0 \\ 0 & D^{-1} \end{matrix}\right] \left[ \begin{matrix} I_p & -BD^{-1} \\ 0 & I_q \end{matrix}\right] \\[12pt] & = \left[ \begin{matrix} \left(A-B D^{-1} C \right)^{-1} & -\left(A-B D^{-1} C \right)^{-1} B D^{-1} \\ -D^{-1}C\left(A-B D^{-1} C \right)^{-1} & D^{-1}+ D^{-1} C \left(A-B D^{-1} C \right)^{-1} B D^{-1} \end{matrix} \right]. \end{align} \)

C.f. matrix inversion lemma which illustrates relationships between the above and the equivalent derivation with the roles of A and D interchanged.

If M is a positive-definite symmetric matrix, then so is the Schur complement of D in M.

If p and q are both 1 (i.e. A, B, C and D are all scalars), we get the familiar formula for the inverse of a 2-by-2 matrix:

\( M^{-1} = \frac{1}{AD-BC} \left[ \begin{matrix} D & -B \\ -C & A \end{matrix}\right] \)

provided that AD − BC is non-zero.
Application to solving linear equations

The Schur complement arises naturally in solving a system of linear equations such as

\( Ax + By = a \, \)
\( Cx + Dy = b \, \)

where x, a are p-dimensional column vectors, y, b are q-dimensional column vectors, and A, B, C, D are as above. Multiplying the bottom equation by \( \( BD^{-1} \) and then subtracting from the top equation one obtains

\( (A - BD^{-1} C) x = a - BD^{-1} b.\, \)

Thus if one can invert D as well as the Schur complement of D, one can solve for x, and then by using the equation Cx + Dy = b one can solve for y. This reduces the problem of inverting a \( (p+q) \times (p+q) \) matrix to that of inverting a p×p matrix and a q×q matrix. In practice one needs D to be well-conditioned in order for this algorithm to be numerically accurate.
Applications to probability theory and statistics

Suppose the random column vectors X, Y live in Rn and Rm respectively, and the vector (X, Y) in Rn+m has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

\( V=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right], \)

where A is n-by-n and C is m-by-m.

Then the conditional variance of X given Y is the Schur complement of C in V:

\( \operatorname{var}(X\mid Y) = A-BC^{-1}B^T. \)

If we take the matrix V above to be, not a variance of a random vector, but a sample variance, then it may have a Wishart distribution. In that case, the Schur complement of C in V also has a Wishart distribution.
Schur complement condition for positive definiteness

Let X be a symmetric matrix given by

\( X=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]. \)

Let S be the Schur complement of A in X, that is:

\( S= C - B^T A^{-1} B . \, \)

Then

X is positive definite if and only if A and S are both positive definite:

\( X \succ 0 \Leftrightarrow A \succ 0, S = C - B^T A^{-1} B \succ 0. \)

X is positive definite if and only if C and \( A - B C^{-1} B^T \) are both positive definite:

\( X \succ 0 \Leftrightarrow C \succ 0, A - B C^{-1} B^T \succ 0. \)

If A is positive definite, then X is positive semidefinite if and only if S is positive semidefinite:

\( \text{If} A \succ 0, \text{then} X \succeq 0 \Leftrightarrow S = C - B^T A^{-1} B \succeq 0. \)

If C is positive definite, then X is positive semidefinite if and only if \( A - B C^{-1} B^T \) is positive semidefinite:

\( \text{If} C \succ 0, \text{then} X \succeq 0 \Leftrightarrow A - B C^{-1} B^T \succeq 0. \)

These statements can be derived[3] by considering the minimizer of the quantity

\( u^T A u + 2 v^T B^T u + v^T C v, \, \)

as a function of u (for fixed v).
See also

Woodbury matrix identity
Quasi-Newton method
Haynsworth inertia additivity formula

References

^ Zhang, Fuzhen (2005). The Schur Complement and Its Applications. Springer. doi:10.1007/b105056. ISBN 0387242716.
^ Haynsworth, E. V., "On the Schur Complement", Basel Mathematical Notes, #BNB 20, 17 pages, June 1968.
^ Boyd, S. and Vandenberghe, L. (2004), "Convex Optimization", Cambridge University Press (Appendix A.5.5)

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