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1969•Proceedings of the American Mathematical SocietyRequires access

An identity for the Schur complement of a matrix

Douglas E. Crabtree, Emilie V. Haynsworth

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Abstract

This result is known as Schur's formula. In case A is Hermitian, Haynsworth [5] has shown that the inertia of A can be determined from the inertia of any nonsingular principal submatrix of A together with that of its Schur complement. Other applications and properties of the Schur complement will appear in a later paper. In ?2 of this note, we prove that the Schur complement can also be constructed using quotients of minors of A. Details on this method of construction and its relation to partitioned matrices and Mmatrices can be found in [1], [2], [3]. In ?3, this construction is used to prove a quotient identity for the Schur complement: (A/B) = ((A/C)/(B/C)).

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What this paper is about

This result is known as Schur's formula. In case A is Hermitian, Haynsworth [5] has shown that the inertia of A can be determined from the inertia of any nonsingular principal submatrix of A together with that of its Schur complement. Other applications and properties of the Schur complement will appear in a later paper. In ?2 of this note, we prove that the Schur complement can also be constructed using quotients of minors of A. Details on this method of construction and its relation to partitioned matrices and Mmatrices can be found in [1], [2], [3]. In ?3, this construction is used to prove a quotient identity for the Schur complement: (A/B) = ((A/C)/(B/C)).

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Available abstract

This result is known as Schur's formula. In case A is Hermitian, Haynsworth [5] has shown that the inertia of A can be determined from the inertia of any nonsingular principal submatrix of A together with that of its Schur complement. Other applications and properties of the Schur complement will appear in a later paper. In ?2 of this note, we prove that the Schur complement can also be constructed using quotients of minors of A. Details on this method of construction and its relation to partitioned matrices and Mmatrices can be found in [1], [2], [3]. In ?3, this construction is used to prove a quotient identity for the Schur complement: (A/B) = ((A/C)/(B/C)).

Key concepts: Schur complement, Schur's theorem, Complement (music), Schur product theorem, Mathematics, Schur algebra, Invertible matrix, Quotient

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