Fundamental solutions and fundamental matrices

[faradayscat]

I'll put you in context for the sake of simplicity before asking my question. Say we have the following homogeneous linear system:

x'=Ax

Let A be 2x2 for simplicity. Then the general solution would look like:

x(t) = αa + βb

And a fundamental matrix would be:

Ψ(t) = ( a , b )

What confuses me is this: I tried making a new fundamental matrix by replacing the first column of Ψ(t) by a linear combination of the general solution, something like:

x(t) = 2a + 4b

Now my new fundamental matrix looked like this:

Ψ(t) = ( 2a + 4b , b )

And expanding the following expression: x(t)=Ψ(t)c, where c is the vector of constants, I found out that I get the same general solution x(t), with different eigenvectors (however they were simply scalar multiples of the eigenvectors of the matrix A)

My question is this, are linear combinations of the fundamental set of solutions also a fundamental set of solutions? Like, would

{ 2a + 4b , b }

also be a fundamental set of solutions? I guess it would because they are linearly independent... If not, why do we call Ψ(t) a fundamental matrix when we can build one using linear combinations of the fundamental set of solutions? All these questions confuse me, I just need some clarification.

Thanks in advance!

 

[adamjts]

I'm actually just learning about this myself. Let me see if I can take a stab at answering this, and hopefully either 1) we can work it out together or 2) someone will correct me.

Here's my take:

Essentially what you have from the fundamental matrix is a collection of column vectors, each of which is an eigenvector of A. So what makes an eigenvector an eigenvector? An eigenspace of a matrix, A, corresponds to a distinct eigenvalue. So when we have, in your case, 2 eigenvectors, there are a couple possibilities.

Case 1)
-They correspond to the same eigenvalue. In this case, then the two eigenvectors should span your eigenspace. In this case, all linear combinations of these two vectors should also be eigenvectors, and your altered fundamental matrix should work out.

Case 2)
-They correspond to different eigenvalues. In this case, the two eigenvectors do not define an eigenspace. So linear combinations of them do not lie in an eigenspace, and therefore your new vector 2a + 4b would not be an eigenvector since a and b are linearly independent. So your new matrix is not a fundamental matrix.


Does this make sense?