Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider real diagonal (known) matrices $A$, $B$, and $C$, and a comforming matrix $\Pi$.

$C + \Pi B + \Pi A \Pi = 0$

I have been trying to solve this system using elementary algebra. I have two questions: (1) Is this a Ricatti equation? ; and (2) does this have an analytical solution?


share|cite|improve this question
up vote 1 down vote accepted

(1) The Nonsymmetric Algebraic Riccati Equation is $$C + \Pi B + D \Pi + \Pi A \Pi = 0$$ The $D$ matrix in your case is zero.

(2) Your situation is much simpler. If each matrix is diagonal, look at $\Pi = diag(x_i)$ as your solution. Then for $A=diag(a_i)$ $B=diag(b_i)$ $C=diag(c_i)$ the matrix equation is easily seperable into the equations: $$c_i + x_ib_i + x_i^2a_i =0$$ And each term is found with the usual quadratic formula. Each equation will have two solutions unless there is a repeated root. So in general, as a system there are $2^n$ different matrices $\Pi$ to solve your formula, the choice of one of two possibilities for each diagonal in $\Pi$.

share|cite|improve this answer
Great thanks. Do you have a reference text to begin studying 'proper'Riccati equations? – Luap Nalehw Oct 25 '12 at 14:19
I only saw anything about it once myself recently (there must be more references), in a pdf titled The Cyclic Reduction Algorithm: From Poisson Equation to Stochastic Processes and Beyond In memoriam of Gene H. Golub – adam W Oct 25 '12 at 15:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.