In matrix theory the Schur complement of a square submatrix A in the bigger square matrix

is defined as . Here we assume A and D are square matrices. Similarly the Schur complement of D in M is . The formulas are reasonably easy to recall: go clockwise around the block matrix from the opposite corner, remembering to invert the opposite square submatrix.

Here are some properties and uses of the Schur complement.

**1.** It comes up in Gaussian elimination in the solution to a system of equations:

To eliminate x in the system of equations Ax + By = u, Cx + Dy = v, we multiply the first equation by and subtract it from the second equation to obtain .

**2.** The Schur determinant formula:

This is proved by taking determinants of both sides of the matrix factorisation

The formula tells us that a large matrix is invertible if and only if any top left or bottom right submatrix and its Schur complement are invertible.

**3.** The Guttman rank additivity formula [1, p14]:

**4.** It arises in matrix block inversion: in particular the 2,2 element of the inverse of the block matrix is the inverse of the Schur complement of A.

**5.** The addition theorem for the Schur complement of Hermitian matrices ([1, p28]):

Firstly we define the inertia of a matrix. Let be the triple (p,q,z) of integers representing the number of positive, negative and zero eigenvalues of the Hermitian matrix M () respectively. Then

.

**6.** It comes up in the minimisation of quadratic forms [3, App A5.5]:

If A is positive definite, the quadratic form in the variable u (v is a constant vector) has minimum value of , achieved when . This follows from the completion of squares:

**7.** A least squares application of **6** is the following:

Let x and y be two random vectors. The covariance of the error in linearly estimating x from y via K (in order to minimise the expectation , assuming the covariance of y is positive definite, is the Schur complement of in the joint covariance matrix

Similarly the covariance of the error in estimating y from x is the Schur complement of in :

.

**8.** The above leads to a matrix version of the Cauchy Schwartz Inequality [4]:

Since any covariance matrix is non-negative definite we have . With the inner product of random vectors in defined as this becomes

#### References

[1] F. Zhang, “The Schur Complement and its applications”, Springer 2005.

[2] “Block matrix decompositions” in https://ccrma.stanford.edu/~jos/lattice/Block_matrix_decompositions.html

[3] Boyd and Vandenburghe, “Convex Optimisation”, Cambridge University Press, 2004.

[4] Kailath, Sayed and Hassibi, “Linear Estimation”, Prentice Hall, 2000.

[5] Schur complement, Wikipedia

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