Trying to understand eigenvalues with respect to differential equations.I am trying to understand how to find eigenvalues from a matrix consisting of exponential terms, considering a differential equation. The examples I've seen online are ODEs. Without using a vector with exponential terms, here is what I have learned.ddtx→(t)=λx→(t)ddt&nbsp;x1(t)&nbsp;x2(t)&nbsp;x3(t)=&nbsp;λ100&nbsp;0λ20&nbsp;00λ3&nbsp;x1(t=0)&nbsp;x2(t=0)&nbsp;x3(t=0)Here is what I am trying to understandSection of a paper with imag eigenvalueIn this paper, the following assumption is made.J→(t)=|J→|e-iωt=|Jx|e-iωt|Jy|e-iωt|Jz|e-iωtThey are using partial derivatives (I believe this can be viewed as an ODE then?). Differentiating with respect to time I believe yields the following. Please correct me if I am wrong.∂∂tJ→(t)=∂∂t|Jx|e-iωt|Jy|e-iωt|Jz|e-iωt=&nbsp;-iω00&nbsp;0-iω0&nbsp;00-iω|Jx|e-iωt(t=0)|Jy|e-iωt(t=0)|Jz|e-iωt(t=0)Or∂∂t|Jx|e-iωt|Jy|e-iωt|Jz|e-iωt=&nbsp;-iω00&nbsp;0-iω0&nbsp;00-iω|Jx||Jy||Jz|Are my assumptions correct? If so, is there a deeper analysis to why this is the case with exponential terms?

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Trying to understand eigenvalues with respect to differential equations.I am trying to understand how to find eigenvalues from a matrix consisting of exponential terms, considering a differential equation. The examples I&#039;ve seen online are ODEs. Without using a vector with exponential terms, here is what I have learned.ddtx→(t)=λx→(t)ddt&amp;nbsp;x1(t)&amp;nbsp;x2(t)&amp;nbsp;x3(t)=&amp;nbsp;λ100&amp;nbsp;0λ20&amp;nbsp;00λ3&amp;nbsp;x1(t=0)&amp;nbsp;x2(t=0)&amp;nbsp;x3(t=0)Here is what I am trying to understandSection of a paper with imag eigenvalueIn this paper, the following assumption is made.J→(t)=|J→|e-iωt=|Jx|e-iωt|Jy|e-iωt|Jz|e-iωtThey are using partial derivatives (I believe this can be viewed as an ODE then?). Differentiating with respect to time I believe yields the following. Please correct me if I am wrong.∂∂tJ→(t)=∂∂t|Jx|e-iωt|Jy|e-iωt|Jz|e-iωt=&amp;nbsp;-iω00&amp;nbsp;0-iω0&amp;nbsp;00-iω|Jx|e-iωt(t=0)|Jy|e-iωt(t=0)|Jz|e-iωt(t=0)Or∂∂t|Jx|e-iωt|Jy|e-iωt|Jz|e-iωt=&amp;nbsp;-iω00&amp;nbsp;0-iω0&amp;nbsp;00-iω|Jx||Jy||Jz|Are my assumptions correct? If so, is there a deeper analysis to why this is the case with exponential terms?

strasnihwhge · Accepted Answer

Step 1Not sure this is what you want to know, but here is a short explanation about the relation between a system of linear ODEs and Eigendecomposition.Assume a system of first order linear ODEs with constant coefficients. It can be written in matrix form:dx→dt=Mx→.Now assume an exponential solution of the formx→=v→ezt.By plugging it in the equation, you getzv→ezt=Mv→eztwhich expresses that z is an Eigenvalue of M, associated to the Eigenvector v→:Mv→=zv→.Step 2So, after finding a diagonalizationM=TΛT−1and setting x→=Ty→the system reduces to dy→dt=Λy→which is a system of three easy independent equationsdykdt=zkyk.The initial conditions can be dealt with by using y→0=T−1x→0.

Trying to understand eigenvalues with respect to differential

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