I'll put you in context for the sake of simplicity before asking my question. Say we have the following homogeneous linear system:(adsbygoogle = window.adsbygoogle || []).push({});

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, wherecis the vector of constants, I found out that I get the same general solutionx(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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Fundamental solutions and fundamental matrices

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**