Verify eigenvalues of a TST matrix

1. May 13, 2015

binbagsss

1. The problem statement, all variables and given/known data

I have $A=TST(-1,2-1),$ and I need to show that an eigenvector of A is,$Y_{j}=sin(kj \pi / J).$
and then find the full set of eigenvalues of A.

The matrix A comes from writing $-U_{j-1}+2U-U_{j+1}=h^{2}f(x_{j}), 1\le j \le J-1$, in the form $AU=b$

2. Relevant equations

The above.

3. The attempt at a solution

Since the TST is a j-1 x j-1 dimensional, I'm unsure how to approach the algebra.

I'm unsure how to get started, and the notation is confusing me too- I know that the eigenvalues of a $TST(\alpha, \beta)$ are $\alpha+2\beta cos(k\pi/m+1)$ where $k=1....m$,

so here do I need to take $k=j$ and $m+1=J$, but this doesn't really make sense to be as in the above, $k$ and $m$ are anything just with the contrainst $k=1....m$, aren't they?

i'm unsure how to construct the eigenvector $Y$, should it be $AY_{j}= A (Y_{0},....,Y_{J})^{T}$,
so $j$ runs from $1$ to $J-1$?

Any help really appreciated ! Thanks alot !

2. May 13, 2015

Ray Vickson

What does TST mean?

3. May 13, 2015

Staff: Mentor

I don't know what TST means either. Don't assume that an acronym you use is understood by all.

4. May 13, 2015