Recent content by Leo321

Ok, I think I solved what I wanted through a different path. Thanks for the attempts, even if they were only at the mental level.

Can z be expressed as a function of x,y? Or is there some function so that z(t)<f(x(t),y(t))? We can assume that x(t)>0,y(t)>0.

I have two systems of linear differential equations: \frac{dx}{dt}=Ax, \frac{dy}{dt}=By x,y are vectors of length n and A,B are nxn matrices. I have a third system defined by: \frac{dz}{dt}=-ABz Is there anything we can say about what the third system represents in terms of the first two? If...

Thanks a lot for all the answers. In the end it seems that we found a way around this issue, so we don't rely on the derivative of eigenvalues at all.

For the 2x2 case, we get\lambda=\frac{a+d}{2}+\frac{\sqrt{4bc+(a-d)^{2}}}{2} The value inside the square root is always positive, and this function seems to be always differentiable. Right? Any ideas about a general nxn matrix?

I tried to make it easier, but maybe it had the opposite effect. I am interested in the largest eigenvalue only. I do know that if the largest eigenvalue occurs more than once, the derivative might not exist. But.. In my case all the elements of A are positive. According to my understanding...

I have a matrix A, which contains only positive real elements. A is a differentiable function of t. Are the eigenvalues of A differentiable by t?

I found the following. For a given matrix A and vectors b,c, this will transform AT into A and also give c = Pb. C=\begin{bmatrix} b & Ab & \dots & A^{n-1}b \end{bmatrix} O=\begin{bmatrix} c & A'c & \dots & (A')^{n-1}c \end{bmatrix} P= O C ^{-1}

No, I need X-1AX=AT.

I have a real nxn matrix A and I want to find P, so that P-1AP=AT. Does such a matrix exist? How do I find it? What if I have two matrices A,B. Does there exist a matrix P, that transforms both of them to their transposes? Thanks

Any ideas?

Thanks!

I try to understand how to calculate derivatives of functions, which contain matrices. For a start I am looking at derivatives by a single variable. I have x=f(t) and I want to calculate \frac{dx}{dt}. The caveat is that f contains matrices, that depend on t. Can I use the ordinary chain rule...

I have an (unknown) matrix A and with real non-negative values. I know its largest eigenvalue \lambda and the associated eigenvector, v. (I know nothing about the other eigenvectors). Does this give me any information about the eigenvector of AT associated with \lambda or is it completely...

The question is what can we know about the product if there are no shared eigenvectors. What happens for example if the eigenvectors are close, but not the same? Is there anything we can say about the eigenvectors of the product based on the eigenvectors and eigenvalues of A and B?