Matrix differential equation for rectangular matrix

[weetabixharry]

Given a matrix differential equation (system of equations?) of the form:

\textbf{X}^{\prime}(t) = \textbf{AX}(t)

(where X is a complex matrix, t is real scalar and A is always a square and normal real matrix) I am able to find (e.g. here) that a general solution for square \textbf{X} is:

\textbf{X}(t) = \textbf{E}diag\{exp\{\underline{\lambda}t\}\}

where \textbf{E} is the matrix whose columns are the eigenvectors of A and \underline{\lambda} the vector of corresponding eigenvalues. diag\{exp\{\underline{\lambda}t\}\} is a diagonal matrix, with diagonal entries exp\{\underline{\lambda}t\}.

However, what do I do if \textbf{X} is a "tall" rectangular matrix? (i.e. X is (MxN), where M>N)? Can I somehow select only N of the eigenvectors/values?

Thanks very much for any help!

 

[pasmith]

Given a matrix differential equation (system of equations?) of the form:

\textbf{X}^{\prime}(t) = \textbf{AX}(t)

(where X is a complex matrix, t is real scalar and A is always a square and normal real matrix) I am able to find (e.g. here) that a general solution for square \textbf{X} is:

\textbf{X}(t) = \textbf{E}diag\{exp\{\underline{\lambda}t\}\}

where \textbf{E} is the matrix whose columns are the eigenvectors of A and \underline{\lambda} the vector of corresponding eigenvalues. diag\{exp\{\underline{\lambda}t\}\} is a diagonal matrix, with diagonal entries exp\{\underline{\lambda}t\}.

That cannot be correct; it does not include an arbitrary constant.

The solution of
X' = AX
where X (and X') is MxN and A is MxM (required for the matrix multiplication to be defined) and constant is
<br /> X(t) = \exp(At)X(0)<br />
where
<br /> \exp(A) = \sum_{n=0}^{\infty} \frac1{n!} A^n.<br />

Now it is true that if A is diagonalizable then one way to calculate \exp(At) is to use the relation A^n = P^{-1}\Lambda^nP, where \Lambda is diagonal, to obtain \exp(At) = P^{-1}\exp(\Lambda t)P. It is then easily shown from the above definition that \exp(\mathrm{diag}(\lambda_1,\dots,\lambda_M)) = \mathrm{diag}(e^{\lambda_1}, \dots, e^{\lambda_M}), so that
<br /> X(t) = P^{-1} \mathrm{diag}(e^{\lambda_1 t}, \dots, e^{\lambda_M t})PX(0)<br />
where, in your notation, E = P^{-1}.

However, what do I do if \textbf{X} is a "tall" rectangular matrix? (i.e. X is (MxN), where M>N)?

This is not a problem; the above solution works whether X is square or not.

 

[grep6]

It is a problem to take the exponential of a non-square matrix.
How can you calculate A^n when you can't multiply a non-square matrix with itself, its non conformable.

 

[pasmith]

It is a problem to take the exponential of a non-square matrix.
How can you calculate A^n when you can't multiply a non-square matrix with itself, its non conformable.

A must be square; otherwise the matrix equation
<br /> X&#039; = AX<br />
does not make sense.