Non-symmetric Riccati Equation

• Non-symmetric Riccati equation is a quadratic matrix equation with a bilinear unknown, defined without the symmetry constraints of classical Riccati theory.
• The topic covers diverse formulations including cone-preserving, M-matrix, T-Riccati, and differential versions, employing methods like Newton–Kleinman and structure-preserving doubling.
• Its applications range from transport theory to forward-backward stochastic control, emphasizing efficient computation, stability analysis, and invariant subspace characterization.

Non-symmetric Riccati equation denotes a family of quadratic matrix equations in which the unknown matrix enters bilinearly, but the coefficient structure does not enforce the symmetry assumptions of the classical symmetric Riccati theory. In the recent cone-theoretic formulation, the nonsymmetric algebraic Riccati equation is

$XBX+DX+XA+C=0,$

and a stabilizing solution is one for which $A_{\rm cl}=A+BX$ and $D_{\rm cl}=D+XB$ are both stable (Vladu et al., 2024). Across transport theory, M-matrix analysis, T-Sylvester generalizations, differential equations, and forward-backward stochastic control, the subject includes minimal nonnegative solutions, cone-preserving solutions, invariant-subspace descriptions, and numerical methods ranging from monotone iteration and Newton–Kleinman procedures to doubling, projection, and low-rank ADI algorithms (Guo, 2012, Benner et al., 2020, Zulfiqar, 25 Apr 2026, Lü et al., 19 Apr 2025).

1. Principal equation classes

The literature uses several closely related nonsymmetric Riccati models. They differ in sign convention, linear operator, and intended application, but all retain a quadratic term in the unknown matrix and a nonsymmetric coefficient structure.

Variant	Equation	Setting
Cone-preserving algebraic Riccati equation	$XBX+DX+XA+C=0$	Proper cones and stability (Vladu et al., 2024)
Transport-theory NARE	$AX+XB-XCX+E=0$	Large-scale transport theory (Guo, 2014)
M-matrix / linearization form	$XCX-XD-AX+B=0$	Transport theory and M-matrix analysis (Chiang et al., 2011, Iannazzo et al., 2010)
Nonsymmetric T-Riccati equation	$DX+X^TA-X^TBX+C=0$	T-Sylvester/T-Lyapunov operator (Benner et al., 2020)
Nonsymmetric differential Riccati equation	$\dot X=-AX-XD+XSX+Q$	Large-scale time-dependent problems (Angelova et al., 2018)
Equilibrium Riccati equations	coupled equations for $P_1(t,s)$ and $P_2(s)$	Forward-backward stochastic LQ control (Lü et al., 19 Apr 2025)

A further large-scale generalized NARE is written as

$A_{\rm cl}=A+BX$0

with $A_{\rm cl}=A+BX$1, low-rank data, and dimensions $A_{\rm cl}=A+BX$2 in the applications emphasized in the low-rank ADI literature (Zulfiqar, 25 Apr 2026).

The T-Riccati equation replaces the standard Sylvester operator by the T-Sylvester operator

$A_{\rm cl}=A+BX$3

and reduces to the T-Lyapunov equation when $A_{\rm cl}=A+BX$4 (Benner et al., 2020). The differential form introduces a time derivative and is treated by projection onto low-dimensional subspaces, after which a reduced Riccati differential equation is solved (Angelova et al., 2018).

2. Order structures, positivity, and solution concepts

A central distinction in nonsymmetric Riccati theory is the ordering used to define admissible solutions. In the M-matrix literature, the main object is the minimal nonnegative solution $A_{\rm cl}=A+BX$5, meaning the entrywise smallest nonnegative solution (Chiang et al., 2011, Guo, 2012). In that framework, the notion of minimality replaces the usual positive-semidefinite ordering of the symmetric theory (Guo, 2012).

The cone-theoretic formulation replaces entrywise nonnegativity by an arbitrary proper cone $A_{\rm cl}=A+BX$6. A proper cone is convex, closed, pointed, and has nonempty interior, and it induces the partial order

$A_{\rm cl}=A+BX$7

A matrix $A_{\rm cl}=A+BX$8 is $A_{\rm cl}=A+BX$9-nonnegative, or cone-preserving, if $D_{\rm cl}=D+XB$0. A solution $D_{\rm cl}=D+XB$1 of the Riccati equation is cone-preserving if $D_{\rm cl}=D+XB$2, where $D_{\rm cl}=D+XB$3 is the cone of linear operators preserving $D_{\rm cl}=D+XB$4 (Vladu et al., 2024). When $D_{\rm cl}=D+XB$5, this reduces to entrywise nonnegativity.

The same framework introduces cross-positivity. If $D_{\rm cl}=D+XB$6 is the dual cone, then $D_{\rm cl}=D+XB$7 is cross-positive on $D_{\rm cl}=D+XB$8 if

$D_{\rm cl}=D+XB$9

Equivalently,

$XBX+DX+XA+C=0$0

On the nonnegative orthant, cross-positivity means that all off-diagonal entries are nonnegative (Vladu et al., 2024).

The T-Riccati setting also uses an order structure, but it is expressed directly through sign assumptions: $XBX+DX+XA+C=0$1, $XBX+DX+XA+C=0$2, and the Kronecker representation of $XBX+DX+XA+C=0$3 is a nonsingular M-matrix. Under these assumptions, fixed-point and Newton–Kleinman sequences are nondecreasing and converge to the minimal nonnegative solution (Benner et al., 2020).

A plausible implication is that entrywise nonnegativity is only one special case of a broader order-theoretic picture. The cone-preserving formulation makes this explicit by extending orthant-based results to arbitrary proper cones (Vladu et al., 2024).

3. Existence theory, stability, and invariant subspaces

One of the sharpest recent existence criteria concerns the block matrix

$XBX+DX+XA+C=0$4

If $XBX+DX+XA+C=0$5 is cross-positive on $XBX+DX+XA+C=0$6 and stable, meaning $XBX+DX+XA+C=0$7 for every eigenvalue $XBX+DX+XA+C=0$8 of $XBX+DX+XA+C=0$9, then the Riccati equation

$AX+XB-XCX+E=0$0

admits a unique cone-preserving solution $AX+XB-XCX+E=0$1 such that $AX+XB-XCX+E=0$2 and $AX+XB-XCX+E=0$3 are both stable and cross-positive on $AX+XB-XCX+E=0$4. Conversely, if such a cone-preserving stabilizing solution exists, then $AX+XB-XCX+E=0$5 must be stable (Vladu et al., 2024). This result extends the nonnegative orthant case and turns a previously sufficient condition into an equivalence.

The M-matrix literature treats a different but related block structure. With

$AX+XB-XCX+E=0$6

the algebraic Riccati equation

$AX+XB-XCX+E=0$7

has a minimal nonnegative solution when $AX+XB-XCX+E=0$8 is a nonsingular M-matrix or an irreducible singular M-matrix. A further extension shows that if $AX+XB-XCX+E=0$9 is a reducible singular M-matrix but regular, in the sense that there exists $XCX-XD-AX+B=0$0 such that $XCX-XD-AX+B=0$1, then the minimal nonnegative solution still exists and is unique (Guo, 2012).

Invariant-subspace formulations are fundamental in both analysis and computation. For the linearizing matrix

$XCX-XD-AX+B=0$2

a matrix $XCX-XD-AX+B=0$3 solves the NARE if and only if

$XCX-XD-AX+B=0$4

so the columns of $XCX-XD-AX+B=0$5 span an invariant subspace of $XCX-XD-AX+B=0$6 (Iannazzo et al., 2010). In the regular M-matrix case, the spectrum of the related linearization splits across the imaginary axis, and the stable eigenvalues coincide with the spectrum of $XCX-XD-AX+B=0$7 (Guo, 2012).

A recurrent difficulty is the critical or close-to-critical regime. In the critical transport-theory case $XCX-XD-AX+B=0$8, the block matrix $XCX-XD-AX+B=0$9 is singular, factors such as $DX+X^TA-X^TBX+C=0$0 may become singular, and doubling iterations may stall or fail (Chiang et al., 2011). The close-to-critical case arises when problematic eigenvalues are small but nonzero, producing ill-conditioning and slow convergence even though the singular critical theory no longer applies directly (Iannazzo et al., 2010).

4. Iterative methods and structure-preserving algorithms

Monotone fixed-point iteration is the oldest common mechanism across the subject. In the cone-preserving algebraic Riccati problem, one uses

$DX+X^TA-X^TBX+C=0$1

obtaining a monotonically increasing sequence of cone-preserving matrices. A key auxiliary result states that such a sequence converges if it is bounded above in a single vectorial direction $DX+X^TA-X^TBX+C=0$2 (Vladu et al., 2024). In the T-Riccati setting, the iteration

$DX+X^TA-X^TBX+C=0$3

is well defined under the M-matrix assumptions and converges to the minimal nonnegative solution when an appropriate nonnegative upper bound exists (Benner et al., 2020).

Newton-type methods provide faster local convergence. For the T-Riccati equation, the Fréchet derivative at $DX+X^TA-X^TBX+C=0$4 is

$DX+X^TA-X^TBX+C=0$5

and the Newton–Kleinman update solves a T-Sylvester equation whose coefficients are shifted by the current iterate. Under the same sign and M-matrix assumptions, the iterates remain nonnegative, nondecreasing, and bounded above by the minimal nonnegative solution, and they converge monotonically to that solution (Benner et al., 2020). For classical M-matrix algebraic Riccati equations, Newton’s method is locally quadratic when the shifted coefficients remain nonsingular (Guo, 2012).

The structure-preserving doubling algorithm (SDA) is the main high-order method in transport theory and M-matrix settings. Starting from a Cayley-transformed initialization, the recurrences

$DX+X^TA-X^TBX+C=0$6

converge quadratically in the nonsingular M-matrix case (Guo, 2014, Chiang et al., 2011). In regular singular M-matrix problems, SDA and ADDA remain well defined and quadratically convergent under the simple-zero-eigenvalue assumption on the linearization (Guo, 2012).

Shift strategies were developed to remove the spectral obstructions behind breakdown and slow convergence. In the critical transport-theory case, one- and two-parameter rank-one shifts move the zero eigenvalues of the block matrix while leaving the minimal nonnegative solution unchanged; the shifted SDA is then guaranteed to converge quadratically with no breakdown (Chiang et al., 2011). The subspace-shift technique generalizes this idea to close-to-critical problems by shifting an entire invariant subspace associated with the problematic eigenvalues. If $DX+X^TA-X^TBX+C=0$7 and $DX+X^TA-X^TBX+C=0$8 span the right and left central invariant subspaces, then

$DX+X^TA-X^TBX+C=0$9

scales the selected eigenvalues by $\dot X=-AX-XD+XSX+Q$0, preserves the antistable subspace, and therefore preserves the minimal Riccati solution while improving the convergence rate of doubling iterations (Iannazzo et al., 2010).

5. Large-scale, low-rank, and projected computation

Large-scale nonsymmetric Riccati problems typically arise when the coefficient matrices are sparse or structured and the solution has low numerical rank. In the transport-theory NARE

$\dot X=-AX-XD+XSX+Q$1

the matrices $\dot X=-AX-XD+XSX+Q$2 and $\dot X=-AX-XD+XSX+Q$3 are symmetric and low-ranked, while $\dot X=-AX-XD+XSX+Q$4 and $\dot X=-AX-XD+XSX+Q$5 are rank-one updates of nonsingular diagonal matrices. This permits $\dot X=-AX-XD+XSX+Q$6 solves with shifted $\dot X=-AX-XD+XSX+Q$7 and $\dot X=-AX-XD+XSX+Q$8 by the Sherman–Morrison–Woodbury formula (Guo, 2014).

A balancing strategy can simplify the large-scale SDA. With the diagonal scaling $\dot X=-AX-XD+XSX+Q$9, the transformed coefficients satisfy

$P_1(t,s)$0

and carefully chosen low-rank initial factors imply

$P_1(t,s)$1

As a consequence, the modified large-scale SDA reduces the flop operations of the original SDA$P_1(t,s)$2 by about one half while retaining $P_1(t,s)$3 complexity and $P_1(t,s)$4 memory per iteration (Guo, 2014).

Projection methods are prominent for nonsymmetric differential Riccati equations. Extended block Arnoldi bases $P_1(t,s)$5 and $P_1(t,s)$6 are built for $P_1(t,s)$7 and $P_1(t,s)$8, and the approximation

$P_1(t,s)$9

is obtained by solving a reduced Riccati differential equation for $P_2(s)$0. The reduced problem may be integrated by matrix exponential formulas, BDF schemes, or a two-stage Rosenbrock method (Angelova et al., 2018). For transport-theory differential problems, the rank-one perturbation structure of $P_2(s)$1 and $P_2(s)$2 again makes the required linear solves cheap via Sherman–Morrison–Woodbury (Angelova et al., 2018).

The most general low-rank large-scale development in the supplied literature is a low-rank ADI method for the generalized NARE

$P_2(s)$3

The approximate solution is stored as

$P_2(s)$4

and each iteration requires two sparse shifted linear solves plus low-rank updates. The residual factors as

$P_2(s)$5

which supports Petrov–Galerkin residual orthogonality and autonomous shift generation through small projected eigenproblems (Zulfiqar, 25 Apr 2026). Numerical results include a benchmark example of order $P_2(s)$6, and the method is presented as covering Lyapunov equations, Sylvester equations, and symmetric Riccati equations as special cases (Zulfiqar, 25 Apr 2026).

6. Applications, variants, and conceptual boundaries

Transport theory is the most persistent application domain in the supplied literature. Both the classical NARE

$P_2(s)$7

and the alternative form

$P_2(s)$8

arise from models with rank-one or low-rank scattering terms and diagonal-plus-rank-one coefficients (Chiang et al., 2011, Guo, 2014). Large-scale numerical work is organized around this structure, which permits efficient shifts, balancing, and low-rank updates.

The subject also extends beyond transport and queueing models. The nonsymmetric differential Riccati equation is stated to arise in control theory, transport theory, applied probability, and others (Angelova et al., 2018). The nonsymmetric T-Riccati equation has applications in macroeconomics and policy dynamics, where the linear part is governed by the T-Sylvester operator rather than the standard Sylvester operator (Benner et al., 2020).

A distinct development appears in forward-backward stochastic linear-quadratic control. There, time-inconsistency invalidates the classical dynamic-programming principle, and equilibrium strategies are characterized by a coupled system of matrix-valued, non-local ordinary differential equations with a non-symmetric structure. The unknowns are $P_2(s)$9 and $A_{\rm cl}=A+BX$00, and the feedback law depends on the combination

$A_{\rm cl}=A+BX$01

Under uniform boundedness, uniform positive-definiteness, and monotonicity assumptions, the equilibrium Riccati system admits a unique global solution and satisfies the a priori bound

$A_{\rm cl}=A+BX$02

(Lü et al., 19 Apr 2025). The same source gives a one-dimensional example in which, without the positivity assumptions, the scalar ERE blows up in finite backward time (Lü et al., 19 Apr 2025).

The relation to the symmetric Riccati equation is therefore subtle. In the forward-backward stochastic setting, if $A_{\rm cl}=A+BX$03, $A_{\rm cl}=A+BX$04, and $A_{\rm cl}=A+BX$05, then $A_{\rm cl}=A+BX$06 and the equilibrium Riccati system reduces to the standard symmetric Riccati ODE (Lü et al., 19 Apr 2025). In the M-matrix setting, the comparison is formulated differently: the nonsymmetric problem has no symmetry such as $A_{\rm cl}=A+BX$07 or $A_{\rm cl}=A+BX$08, and minimality is interpreted componentwise rather than through positive-semidefinite order (Guo, 2012). This suggests that the nonsymmetric theory is not a single perturbation of the symmetric case, but a collection of structurally different Riccati problems linked by quadratic matrix nonlinearity, order-preserving mechanisms, and invariant-subspace geometry.