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Matrix differential equation for rectangular matrix 
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#1
Jan913, 03:20 PM

P: 108

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! 


#2
Jan1313, 12:45 PM

HW Helper
Thanks
P: 943

The solution of [tex]X' = AX[/tex] where X (and X') is MxN and A is MxM (required for the matrix multiplication to be defined) and constant is [tex] X(t) = \exp(At)X(0) [/tex] where [tex] \exp(A) = \sum_{n=0}^{\infty} \frac1{n!} A^n. [/tex] Now it is true that if [itex]A[/itex] is diagonalizable then one way to calculate [itex]\exp(At)[/itex] is to use the relation [itex]A^n = P^{1}\Lambda^nP[/itex], where [itex]\Lambda[/itex] is diagonal, to obtain [itex]\exp(At) = P^{1}\exp(\Lambda t)P[/itex]. It is then easily shown from the above definition that [itex]\exp(\mathrm{diag}(\lambda_1,\dots,\lambda_M)) = \mathrm{diag}(e^{\lambda_1}, \dots, e^{\lambda_M})[/itex], so that [tex] X(t) = P^{1} \mathrm{diag}(e^{\lambda_1 t}, \dots, e^{\lambda_M t})PX(0) [/tex] where, in your notation, [itex]E = P^{1}[/itex]. 


#3
Jan1513, 03:16 PM

P: 5

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


#4
Jan1613, 11:10 AM

HW Helper
Thanks
P: 943

Matrix differential equation for rectangular matrix
[tex] X' = AX [/tex] does not make sense. 


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