# Matrix differential equation for rectangular matrix

• weetabixharry
In summary: Therefore, the solution given above is only valid for square X. For a rectangular X, a different approach would need to be taken.
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!

Last edited:
weetabixharry said:
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 can't multiply a non-square matrix with itself, its non conformable.

grep6 said:
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
$$X' = AX$$
does not make sense.

I would like to provide a response to your question regarding the matrix differential equation for rectangular matrix. Firstly, I would like to clarify that the term "rectangular matrix" is not commonly used in mathematics, and it would be more accurate to refer to it as a non-square matrix.

Moving on to your question, the solution for a non-square matrix \textbf{X}(t) would be similar to the one for a square matrix, however, the matrix \textbf{E} would now be an (MxN) matrix, where each column represents an eigenvector of A. In this case, the diagonal matrix diag\{exp\{\underline{\lambda}t\}\} would be of size (NxN) since we are only considering N of the eigenvectors/values.

In order to select only N of the eigenvectors/values, we can use a technique called "truncation", where we only consider the N largest eigenvalues and their corresponding eigenvectors. This can be done by sorting the eigenvalues in descending order and selecting the first N eigenvalues and their eigenvectors.

I hope this helps in providing a solution for your problem. However, I would suggest consulting a mathematician or a numerical analyst for a more detailed and accurate explanation.

## 1. What is a matrix differential equation for rectangular matrix?

A matrix differential equation for a rectangular matrix is a type of differential equation that involves matrices and their derivatives. It is used to model systems that involve multiple variables and their rates of change over time.

## 2. How is a matrix differential equation for rectangular matrix different from a regular differential equation?

A regular differential equation involves only scalar variables and their derivatives, while a matrix differential equation involves matrices and their derivatives. This allows for a more complex and comprehensive representation of a system.

## 3. What are some real-world applications of matrix differential equations for rectangular matrix?

Matrix differential equations for rectangular matrix are commonly used in fields such as physics, engineering, economics, and biology to model and analyze systems that involve multiple variables and their rates of change, such as population growth, chemical reactions, and electrical circuits.

## 4. What techniques are used to solve matrix differential equations for rectangular matrix?

There are various techniques that can be used to solve matrix differential equations for rectangular matrix, such as separation of variables, variation of parameters, and matrix exponential techniques. The specific technique used will depend on the structure and complexity of the equation.

## 5. How are matrix differential equations for rectangular matrix used in machine learning and data analysis?

In machine learning and data analysis, matrix differential equations for rectangular matrix are used to model and analyze complex datasets that involve multiple variables and their relationships. This allows for the development of predictive models and trend analysis to make informed decisions based on data.

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