Thread: 4x4 Complex repeated Eigenvalues

I'm trying to find the general solution for the following 4x4 matrix:
[0 1 1 0]
[-1 0 0 1]
[0 0 0 1]
[0 0 -1 0]

So far, I have the eigenvalues as repeated i, i, - i ,-i
Eigenvector for i: [-i 1 0 0]^t (with multiplicity 2)
Eigenvector for -i: [1 i 0 0]^t (with multiplicity 2)

How do I get the general solution for this? And how do I find the adjoint eigenvectors for a 4x4 with complex repeated eigenvalues?

Originally Posted by Borkborkmath

So far, I have the eigenvalues as repeated i, i, - i ,-i

Right.

How do I get the general solution for this?

What do you mean by general solution?. General solution of ...

Of the matrix,
Umm.. I do believe it should look something like x(t) = c_1e^(eigen)(eigenvector) + ... + c_ne^(eigen)(eigenvector)

This is from an ODE's class, but last time I posted a matrix question from ODE's a mod moved it and scolded me. So, that is why I put it in the linear algebra this time.

I guess you mean the general solution of the differential system . You can express




or in a vectorial form:



where are de columns of .

Yes, thats exactly what I mean.

How do I get the adjoint vectors of the repeated complex eigenvalues so that I can write the general solutions for X'= AX?

Originally Posted by Borkborkmath

Yes, thats exactly what I mean. How do I get the adjoint vectors of the repeated complex eigenvalues so that I can write the general solutions for X'= AX?

Don't worry about it, all is included in the columns of .

So the general solution is:
X(t) = e^(tA)*(i 1 0 0) + e^(tA)*(1 i 0 0)?