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weetabixharry

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

[itex]\textbf{X}^{\prime}(t) = \textbf{AX}(t)[/itex]

(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 [itex]\textbf{X}[/itex] is:

[itex]\textbf{X}(t) = \textbf{E}diag\{exp\{\underline{\lambda}t\}\}[/itex]

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

However, what do I do if [itex]\textbf{X}[/itex] 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!

[itex]\textbf{X}^{\prime}(t) = \textbf{AX}(t)[/itex]

(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 [itex]\textbf{X}[/itex] is:

[itex]\textbf{X}(t) = \textbf{E}diag\{exp\{\underline{\lambda}t\}\}[/itex]

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

However, what do I do if [itex]\textbf{X}[/itex] 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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