Solve a nonlinear matrix equation

[rehan_eme]

TL;DR Summary

Solve a nonlinear matrix equation

Hi all,

I want to know if a second solution exists for the following math equation:

Ce^{At} ρ_p+(CA)^{−1} (e^{At}−I)B=0

Where C, ρ_p, A and B are constant matrices, 't' is scalar variable. I know that atleast one solution i.e. 〖t=θ〗_1 exists, but I want a method to determine if there is another θ_0<θ_1 that is also a (second) solution. Any anlaytical way of determining that is what I am looking for.

 

[hutchphd]

Latex please

 

[fresh_42]

Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/

 

[Vanadium 50]

Add my vote to the hope that LaTex might add some neaded clarity. What does a scalar to a power of a matrix even mean?

 

[renormalize]

What does a scalar to a power of a matrix even mean?

Isn't the matrix exponential (https://en.wikipedia.org/wiki/Matrix_exponential) one well-known example?

 

[DaveE]

What does a scalar to a power of a matrix even mean?

https://en.wikipedia.org/wiki/Matrix_exponential

IRL you do it by transforming it to a diagonal form with eigenvalues so you don't have infinite sums.

 

[Mayhem]

https://en.wikipedia.org/wiki/Matrix_exponential

IRL you do it by transforming it to a diagonal form with eigenvalues so you don't have infinite sums.

Doesn't this show up in QM quite often? I remember seeing something to the effect of $e^{\mathbf{X}}$ before.

 

[fresh_42]

Doesn't this show up in QM quite often? I remember seeing something to the effect of $e^{\mathbf{X}}$ before.

The exponentiation comes into play whenever we want to solve a differential equation when we turn something linear into something curved, the transition from a Lie algebra to a Lie group. However, the OP didn't provide any such context, no differential equation, no vector field. It is not even clear whether $e^{At}$ converges. It looks like a flow, but it would have been nice to know for sure.

 

[Vanadium 50]

I have not seen it in QM, or for that matter, ever. Raising a scalar to the identity matrix returns what? (Apparently, a disgonal matrix with the scalar on the diagonal,,,guess it has to be something)

 

[fresh_42]

I have not seen it in QM, or for that matter, ever. Raising a scalar to the identity matrix returns what? (Apparently, a disgonal matrix with the scalar on the diagonal,,,guess it has to be something)

Here is what I think it should have been:

TL;DR Summary: Solve a nonlinear matrix equation

Hi all,

I want to know if a second solution exists for the following math equation:

$Ce^{At} ρ_p + (CA)^{−1} (e^{At}−I) B=0$ or $C\exp(At) ρ_p + (CA)^{−1} (\exp(At) −I ) B=0$

Where $C, ρ_p, A,$ and $B$ are constant matrices, $t$ is a scalar variable. I know that at least one solution i.e. $t=\theta_1$ exists, but I want a method to determine if there is another $\theta_0<\theta_1$ that is also a (second) solution. Any analytical way of determining that is what I am looking for.

E.g. it could be that $A\in \mathfrak{su}(2)$ and $e^{At}$ is a flow in $\operatorname{SU}(2).$

 

[Mark44]

Raising a scalar to the identity matrix returns what? (Apparently, a disgonal matrix with the scalar on the diagonal,,,guess it has to be something)

This is something that appears in all but the most elementary linear algebra textbooks. As a Maclaurin expansion $e^A = I + A + \frac {A^2}{2!} + \frac {A^3}{3!} + \dots + \frac {A^n}{n!} + \dots$.

$e^I = I + I + \frac {I^2}{2!} + \frac {I^3}{3!} + \dots + \frac {I^n}{n!} + \dots$
$= I(2 + 1/2 + 1/6 + 1/24 + \dots) = eI$

 

[renormalize]

I have not seen it in QM, or for that matter, ever.

That's surprising to me. Maybe I'm showing my age here, but back in the day, quantum mechanics textbooks often exponentiated the time-independent, Hermetian Hamiltonian $H$ (a differential operator or a matrix, depending on the representation) of a quantum system to define the unitary operator $T$ of time evolution (e.g., see the page from Merzbacher below). Maybe the QM texts you've studied no longer use that approach?

 

[DaveE]

This is routinely used as the canonical solution to a system of linear DEs, state space systems, for example.
https://web.mit.edu/2.14/www/Handouts/StateSpaceResponse.pdf