Matrices and eigenvalues. A comment in my answer.

[sphlanx]

Homework Statement

Hello and thanks again to anyone who has replied my posts. Your help is a great deal and really appreciated.

I have the following homework question which I have answered and I want a comment if it is valid or illogical:

We are given a matrix, with eigenvalues 3 and 7 respectively.

We are asked to say if there other matrices with the same eigenvalues and if the set containing all these matrices is finite.

Homework Equations

the matrix given:

2 1
-5 8

The Attempt at a Solution

I have thought of the following answer: My point is that there are infinite matrix with the same eigenvalues that have different eigenvectors. So I say that the linear system:

A(x1,y1)=λ1(x1,y1)
A(x2,y2)=λ2(x2,y2)

where A is a 2x2, has infinite solutions IF we take a11,a12,a21,a22(the elements of the matrix) as the variables of the linear system. The solutions will have x1,y1 and x2,y2 as constant parameters.

Have i got something terribly wrong?

 

[HallsofIvy]

No, that's perfectly correct.

Another way to prove that is to note that the matrix
$$\begin{bmatrix}3 & a \\ 0 & 7\end{bmatrix}$$
has eigenvalues 3 and 7 for any a.