Hello Daniel: which of your playlists does this video belong to? like if I wanted to watch the previous lectures that led up to this particular lecture, where do I look?
Hello Daniel: Thank you for the helpful video on solving a system of differential equations by matrix exponential. Can u give me an idea, on how to find a matrix D (to be found) such that: e^(D ) = C (given square matrix) [Example: for D=(a,b;c,d) such that d11=a,d12=b,d21=c and d22=d and also, where C=(-1,0;0,-4) such that c11=-1,c12=0,c21=0 and c22=-4 which is a diagonal matrix], how to find Matrix D. If I apply ln it will not work as ln(-1) is not defined as D should be real 2 by 2 matrix. Thanks
I can't say for sure, because this is beyond my knowledge, but if you look at: dieci.math.gatech.edu/preps/LucaRealLog.pdf theorem 1.2 says you need even number of Jordan blocks for every non-negative eigenvalues. Your matrix only has one -1 and one -4 so it's not possible. A 4x4 diagonal matrix whose diagonal entries are -1,-1,-4,-4 will have a real log. The -1, -1 can be produced by 2x2 matrix M=[[0,-pi],[pi,0]] while the -4,-4 can be produced by N=M*log(4). The block diagonal 4x4 matrix with diagonal matrix M,N will be the log in that case.
Hello Daniel: which of your playlists does this video belong to? like if I wanted to watch the previous lectures that led up to this particular lecture, where do I look?
Thanks for your kindness.
Thank you for the lesson
Hello Daniel: Thank you for the helpful video on solving a system of differential equations by matrix exponential. Can u give me an idea, on how to find a matrix D (to be found) such that: e^(D ) = C (given square matrix) [Example: for D=(a,b;c,d) such that d11=a,d12=b,d21=c and d22=d and also, where C=(-1,0;0,-4) such that c11=-1,c12=0,c21=0 and c22=-4 which is a diagonal matrix], how to find Matrix D. If I apply ln it will not work as ln(-1) is not defined as D should be real 2 by 2 matrix. Thanks
I can't say for sure, because this is beyond my knowledge, but if you look at:
dieci.math.gatech.edu/preps/LucaRealLog.pdf
theorem 1.2 says you need even number of Jordan blocks for every non-negative eigenvalues. Your matrix only has one -1 and one -4 so it's not possible.
A 4x4 diagonal matrix whose diagonal entries are -1,-1,-4,-4 will have a real log. The -1, -1 can be produced by 2x2 matrix M=[[0,-pi],[pi,0]] while the -4,-4 can be produced by N=M*log(4). The block diagonal 4x4 matrix with diagonal matrix M,N will be the log in that case.
@@daniel_an Thank you very much. I will reach out to the paper you have shared with me. many many thanks.
Danke Schon !