Verify eigenvalues of a TST matrix

[binbagsss]

Homework Statement

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##

Homework Equations

[/B]
The above.

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 a lot !

 

[Ray Vickson]

Homework Statement

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##

Homework Equations

[/B]
The above.

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 a lot !

What does TST mean?

 

[Mark44]

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

 

[binbagsss]

Apologies. http://www.damtp.cam.ac.uk/user/cbs31/Teaching_files/c06.pdf - a Toeplitz symmetric tridiagonal, page 1.