Schur complement

In linear algebra and the theory of matrices,the Schur complement of a block of a matrix within alarger 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

:

so that "M" is a ("p"+"q")×("p"+"q") matrix.

Then the Schur complement of the block "D" of thematrix "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 [citebook |title=The Schur Complement and Its Applications |first=Fuzhen |last=Zhang |year= |publisher=Springer |year=2005|isbn=0387242716 ] .

Background

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

:

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

:

The inverse of "M" thus may be expressed involving $D^\{-1\}$ and the inverse of Schur's complement (if it exists) only as

:::::

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:

:

provided that the determinant $AD-BC$ 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 ofinverting a $(p+q)\; imes\; (p+q)$ 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 R^"n" and R^"m" respectively, and the vector ("X", "Y") in R^"n"+"m" has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

:

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

:$operatorname\{var\}(Xmid\; Y)\; =\; A-BC^\{-1\}B^T.$

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.

References

See also

* Woodbury matrix identity