Page 129 - Matrix Analysis & Applied Linear Algebra

3.7 Matrix Inversion 123

A r×r C r×s
3.7.11. Consider the block matrix . When the indicated in-
R s×r B s×s
verses exist, the matrices defined by
S = B − RA −1 C and T = A − CB −1 R

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are called the Schur complements of A and B, respectively.
(a) If A and S are both nonsingular, verify that

−1 −1 −1 −1 −1 −1 −1
A C A + A CS RA −A CS
= .
R B −S −1 RA −1 S −1
(b) If B and T are nonsingular, verify that

−1 −1 −1 −1
A C T −T CB
= −1 −1 −1 −1 −1 −1 .
R B −B RT B + B RT CB

3.7.12. Suppose that A, B, C, and D are n × n matrices such that AB T
T
T
and CD T are each symmetric and AD − BC = I. Prove that
T
T
A D − C B = I.

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This is named in honor of the German mathematician Issai Schur (1875–1941),who first studied
matrices of this type. Schur was a student and collaborator of Ferdinand Georg Frobenius
(p. 662). Schur and Frobenius were among the first to study matrix theory as a discipline
unto itself,and each made great contributions to the subject. It was Emilie V. Haynsworth
(1916–1987)—a mathematical granddaughter of Schur—who introduced the phrase “Schur
complement” and developed several important aspects of the concept.