- #1
binbagsss
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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 !