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.