Solving matrix equations?

There is a matrix equation A1-A2X+XA3-XA4X=0. where, A1,A2,A3,A4 all are [2X2] matrix. Then how to find X which is also a [2X2] matrix?

[Bruno Luong]

Is A4 is symmetric/hermitian ?

Answer 1

[Matt J]

You can try fsolve, but you will need a good initial guess, X0.

X=fsolve(@(X) A1-A2*X+X*A3-X*A4*X,  X0)

[Torsten]

Any classification of the equation possible (e.g. Riccati) ?

I guess: no, but perhaps someone else has more experience.

[Sam Chak]

It does look like an algebraic Riccati equation to me. It is usually encountered in the time-invariant Linear-Quadratic Regulator (LQR) problem.

https://en.wikipedia.org/wiki/Linear%E2%80%93quadratic_regulator

Conventionally, it can be solved through the matrix decomposition procedure called Schur, named after a Russian mathematician.

https://www.mathworks.com/help/matlab/ref/schur.html

Can also try solving the matrix equation using the LMI Solver (a very powerful and versatile tool for solving a number of problems). But the formulation part can be quite tricky and frustrating is the matrices are very large.

https://www.mathworks.com/help/robust/lmis.html

[Torsten]

Since A2 is not equal to -A3.', I couldn't bring it into the Riccati form MATLAB's idare expects.

Do you have a suggestion ?

[Sam Chak]

Ah! I have forgotten there were care and dare in the olden days.

Let's wait for @Kallol Chatterjee to clarify if .

[Kallol Chatterjee]

The Problem is not a Riccati equation type. Actually on the system equation there is no condition of symetricity. I am converting the system into slow and fast response. For that I need to change the coordinate transformation. To solve this transformation I need this solution.

[Kallol Chatterjee]

There is no relation between A2 and A3.

[Kallol Chatterjee]

@Matt J Thanks for helping me. It finds the solution.