Verify eigenvalues of a TST matrix

In summary, the conversation discusses a problem involving a matrix A that is derived from a tridiagonal system of equations. The task is to show that a certain vector Y is an eigenvector of A and to find the full set of eigenvalues of A. The conversation also includes a discussion about the notation used and how to approach the problem.
  • #1
binbagsss
1,326
12

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 !
 
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  • #2
binbagsss said:

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?
 
  • #3
I don't know what TST means either. Don't assume that an acronym you use is understood by all.
 

FAQ: Verify eigenvalues of a TST matrix

What is a TST matrix?

A TST (transition state theory) matrix is a representation of the transition state of a chemical reaction, which is the point at which the reactants have reached their highest energy and are about to form products.

How do you verify the eigenvalues of a TST matrix?

The eigenvalues of a TST matrix can be verified by diagonalizing the matrix and solving for the eigenvalues. The resulting eigenvalues should be real and positive, as they represent the energy levels of the transition state.

Why is it important to verify the eigenvalues of a TST matrix?

Verifying the eigenvalues of a TST matrix is important because it confirms that the matrix accurately represents the transition state of a chemical reaction. This is crucial for understanding the kinetics and thermodynamics of reactions and designing effective catalysts.

What factors can affect the accuracy of the eigenvalues in a TST matrix?

The accuracy of the eigenvalues in a TST matrix can be affected by the level of theory used to calculate them, the basis set used, and the quality of the potential energy surface. Additionally, the presence of solvent or other environmental effects can also impact the accuracy.

Are there any limitations to using a TST matrix to determine eigenvalues?

Yes, there are limitations to using a TST matrix to determine eigenvalues. The TST model assumes a one-dimensional reaction coordinate, which may not accurately represent more complex reactions. Additionally, the model does not account for quantum effects or tunneling, which can play a significant role in certain reactions.

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