We first present an extension of a recent characterisation of diagonal Riccati stability and, using this, extend a result of Kraaijevanger on diagonal Lyapunov stability to Riccati stability of time-delay systems. We also describe a class of transformations that preserve the property of being diagonally Riccati stable and apply these two results to provide novel stability results for classes of time-delay systems.Keywords: Riccati inequality; diagonal stability; time-delay systems.AMS subject classifications: 15A24; 93D05. Background and IntroductionThe problem of Riccati stability was introduced in [1] and is motivated by the stability theory of linear time-delay systems. Formally, a pair (A, B) is said to be Riccati stable if there exist P = P T ≻ 0, Q = Q T ≻ 0 such thatwhere M ≺ 0 (M ≻ 0) denotes that the matrix M = M T is negative definite (positive definite). Throughout the paper, M 0 (M 0) denotes that M is positive semi-definite (negative semidefinite). When matrices P , Q satisfying (1) exist they define a quadratic Lyapunov-Krasovskii functional establishing stability for the time-delay systeṁwhere τ ≥ 0 can be any fixed nonnegative delay. A ∈ R n×n is Metzler if a ij ≥ 0 for i = j. We denote the spectrum of A by σ(A) and the spectral abscissa of A by µ(A): formally,and say that A is Hurwitz if µ(A) < 0.We denote the standard basis of R n by e 1 , . . . , e n and we use 1 n to denote the vector in R n , all of whose entries are equal to one. For A ∈ R n×n , diag(A) is the vector v in R n with v i = a ii , 1 ≤ i ≤ n. Sym(n, R) denotes the space of n × n symmetric matrices with real entries.For a real number x, sign(x) is given by +1 if x ≥ 0 and −1 for x < 0 respectively.

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