# Matrix differential equation for rectangular matrix

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!

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## Answers and Replies

pasmith
Homework Helper
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
$$X(t) = \exp(At)X(0)$$
where
$$\exp(A) = \sum_{n=0}^{\infty} \frac1{n!} A^n.$$

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
$$X(t) = P^{-1} \mathrm{diag}(e^{\lambda_1 t}, \dots, e^{\lambda_M t})PX(0)$$
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.

It is a problem to take the exponential of a non-square matrix.
How can you calculate A^n when you cant multiply a non-square matrix with itself, its non conformable.

pasmith
Homework Helper
It is a problem to take the exponential of a non-square matrix.
How can you calculate A^n when you cant multiply a non-square matrix with itself, its non conformable.

A must be square; otherwise the matrix equation
$$X' = AX$$
does not make sense.