In mathematics, the Haynsworth inertia additivity formula, discovered by

Emilie Virginia Haynsworth Emilie Virginia Haynsworth (June 1, 1916 – May 4, 1985) was an American mathematician at Auburn University who worked in linear algebra and matrix theory. She gave the name to Schur complements and is the namesake of the Haynsworth inertia ad ...

(1916–1985), concerns the number of positive, negative, and zero

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s of a

Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...

and of block matrices into which it is partitioned. The ''inertia'' of a Hermitian matrix ''H'' is defined as the ordered triple :

\mathrm(H) = \left( \pi(H), \nu(H), \delta(H) \right)

whose components are respectively the numbers of positive, negative, and zero eigenvalues of ''H''. Haynsworth considered a partitioned Hermitian matrix :

H = \begin H_ & H_ \\  H_^\ast & H_ \end

where ''H''₁₁ is

nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...

and ''H''₁₂^* is the

conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...

of ''H''₁₂. The formula states: :

\mathrm \begin H_ & H_ \\  H_^\ast & H_ \end = \mathrm(H_) + \mathrm(H/H_)

where ''H''/''H''₁₁ is the

Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...

of ''H''₁₁ in ''H'': :

H/H_ = H_ - H_^\ast H_^H_.

Generalization

If ''H''₁₁ is

singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar ...

, we can still define the generalized Schur complement, using the

Moore–Penrose inverse In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roge ...

H_^+

instead of

H_^

. The formula does not hold if ''H''₁₁ is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham, to the effect that

\pi(H) \ge \pi(H_) + \pi(H/H_)

and

\nu(H) \ge \nu(H_) + \nu(H/H_)

. Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold.

Notes and references

{{reflist Linear algebra Matrix theory Theorems in algebra

Generalization

See also

Notes and references