Solutions of first-order matrix differential equations

Haorong Wu
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Homework Statement
How to solve the following matrix differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##, where ##A(t)## and ##B(t)## are ##n\times n## matrices and ##\left | \psi \right >## is a ##n##-vector.
Relevant Equations
None
Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.

I suppose I could write the equation as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ## with general solutions being ##\left | \psi \right >=\exp \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ## and ##\left | c\right > ## is a constant vector. Then a first-order approximation may be ##\left | \psi \right >=\left (I+ \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ##.

I am not familiar with matrix differential equations. Does this method have any restrictions or problems? Or there may be other better approximation solutions? Any references would be greatly appreciated.

Thanks!
 
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Haorong Wu said:
Homework Statement:: How to solve the following matrix differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##, where ##A(t)## and ##B(t)## are ##n\times n## matrices and ##\left | \psi \right >## is a ##n##-vector.
Relevant Equations:: None

Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.

I suppose I could write the equation as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ## with general solutions being ##\left | \psi \right >=\exp \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ## and ##\left | c\right > ## is a constant vector. Then a first-order approximation may be ##\left | \psi \right >=\left (I+ \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ##.

I am not familiar with matrix differential equations. Does this method have any restrictions or problems? Or there may be other better approximation solutions? Any references would be greatly appreciated.

Thanks!
I don't understand what you did. Starting from as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ##, wouldn't you get ##\left | \psi \right >= \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) ##? You might be mixing up the concept of an integrating factor with the solution of a differential equation. It's been nearly 25 years since I worked on this stuff, so I could be mistaken.

Nit: Also, you have used the symbol ##\partial_t##. Since the matrices and vector are functions of a single variable t, the ordinary derivative would be more suitable, IMO.
 
Thanks, @Mark44. There is a ##\left | \psi \right >## in the rhs, so it is like the equation as ##y'=\alpha y##, which solution is exponential functions. Also, thanks for the suggestions about ##\partial_t##, but I use ##\partial_t## for partial and ordinary derivatives if no confusion could occur.
 
Haorong Wu said:
Thanks, @Mark44. There is a ##\left | \psi \right >## in the rhs, so it is like the equation as ##y'=\alpha y##, which solution is exponential functions.
OK, I understand. I'm not so familiar with the notation ##|\psi>##, as that's probably more of a physics notation rather than one used in mathematics.
 
Mark44 said:
OK, I understand. I'm not so familiar with the notation ##|\psi>##, as that's probably more of a physics notation rather than one used in mathematics.
Sorry for the confusion. It can be treated as a vector.
 
Assuming B is invertible, we can rewrite the ODE as <br /> \frac{d}{dt}(B\psi) + \left[ AB^{-1} - \frac{dB}{dt}B^{-1}\right](B\psi) = 0 which is of the standard form <br /> \frac{du}{dt} + C(t)u = 0. But in general there is no closed form solution unless C commutes with \int_0^t C(s)\,ds, when the solution is u(t) = \exp\left(\int_0^t C(s)\,ds\right)u(0).
 
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