The wonderful world of matrices is rich in structure, ranging from simple and obvious
element patterns to special properties, all of which can be exploited to develop faster and
more accurate algorithms with lower computational cost and memory requirements than
general methods, as well as to compute more physically meaningful solutions. Arguably,
one of the most desirable structures a matrix can possess is positive (semi)definiteness
and in this talk we present two applications where it plays a major role.
The first is focused on the Bartels-Stewart method to solve stable, non-negative defi-
nite Lyapunov equations \(A^*X + XA = -C\), for which the solution \(X\) should be positive
semidefinite. However, for some problems in practice it happens that the computed so-
lution is in fact indefinite. To overcome this, the general method is modied to compute
the Cholesky factor \(R\) of the solution directly, thus guaranteeing positive semidefiniteness
via \(X = R^TR\).
In the second part of the talk we describe shrinking, a new approach to replacing a
real indefinite matrix with a positive semidefinite one, while the leading positive definite
block of the matrix remains fixed and all the unfixed elements are minimally perturbed.
The problems of this type appear in many financial and data analysis applications. We
show how the problem can be solved by the bisection method and posed as a generalized
eigenvalue problem, and we demonstrate how exploiting positive definiteness in these two
methods leads to impressive computational savings.
 S. J. Hammarling. Numerical Solution of the Stable, Non-negative Definite Lya-
punov Equation. IMA J. Numer. Anal., 2:303-323, 1982
 S. J. Hammarling. Numerical Solution of the Discrete-time, Convergent, Non-
negative Definite Lyapunov Equation. Systems and Control Letters, 17:137-139, 1991
 N. J. Higham, N. Strabic and V. Sego. Restoring Definiteness via Shrinking, with
an Application to Correlation Matrices with a Fixed Block. MIMS EPrint 2014.54