What is a Right Hand Side Vector in Eigenvector Calculations?

[thunderbird]

Homework Statement

I've done part A, and part D is easy. I'm stuck with part B. I have no idea what a "right hand side vector" is...

Homework Equations

The Attempt at a Solution

Part A: All eigenvectors are valid. Eigenvalues are 1, 1/2 and 1/3.

 

[Mark44]

Homework Statement

I've done part A, and part D is easy. I'm stuck with part B. I have no idea what a "right hand side vector" is...

Write each of the vectors e₁, e₂, and e₃ as a linear combination of k₁, k₂, and k₃.

The idea is to write Aⁿe₁ as Aⁿ(c₁k₁ + c₂k₂ + c₃k₃).

Homework Equations

The Attempt at a Solution

Part A: All eigenvectors are valid. Eigenvalues are 1, 1/2 and 1/3.

 

[thunderbird]

How do I find the constants and the e vectors? I don't think I really understand what what I'm supposed to do represents...

 

[Mark44]

The e_i are just the standard basis vectors for R³.
e₁ = <1, 0, 0>^T
e₂ = <0, 1, 0>^T
e₃ = <0, 0, 1>^T

You must have been taught how to write a vector (such as k₁) as a linear combination of other vectors. Check your book and/or notes.

 

[thunderbird]

I looked but it doesn't really say. We don't have a textbook for this actually and the notes are very vague. This is what it says:

We can write
x(0) = c1 k1 + c2 k2 + c3 k3 + c4 k4 (6.2)
for some coefficients c1 , c2 , c3 and c4 uniquely determined. Equation (6.2) can
be written in matrix-vector form
T c = x(0) (6.3)
where c = (c1 , c2 , c3 , c4 )T and T is the 4
× 4 matrix with eigenvectors k1 , k2 , k3
and k4 in its columns. Solving (6.3) (I used MATLAB) gives c1 = 1/2, c2 = 1/2,
c3
≈ −0.6830 and c4 ≈ 0.1830. With these values of c (6.2) is a representation
of x(0) as a linear combination of eigenvectors of P .

 

[Mark44]

I looked but it doesn't really say. We don't have a textbook for this actually and the notes are very vague. This is what it says:

We can write
x(0) = c1 k1 + c2 k2 + c3 k3 + c4 k4 (6.2)
for some coefficients c1 , c2 , c3 and c4 uniquely determined. Equation (6.2) can
be written in matrix-vector form
T c = x(0) (6.3)
where c = (c1 , c2 , c3 , c4 )T and T is the 4
× 4 matrix with eigenvectors k1 , k2 , k3
and k4 in its columns. Solving (6.3) (I used MATLAB) gives c1 = 1/2, c2 = 1/2,
c3
≈ −0.6830 and c4 ≈ 0.1830. With these values of c (6.2) is a representation
of x(0) as a linear combination of eigenvectors of P .

Put this in terms of what you have. You want to find constants c1, c2, c3 so that
e₁ = c₁ k₁ + c₂ k₂ + c₃ k₃

and the same thing (with different sets of constants) for e₂ and e₃.

Then you can calculate Aⁿe₁ = Aⁿ(c₁ k₁ + c₂ k₂ + c₃ k₃).
This works out to c₁ Aⁿ k₁ + c₂ Aⁿ k₂ + c₃ Aⁿ k₃.

Since the k_i's are eigenvectors, you can calculate A k_i, and hopefully you have learned something about the eigenvalues of Aⁿ.

 

[thunderbird]

Ok like this?:

e1 = k1 - k2 + k3
e2 = 2 k1 + 0 k2 + k3
e3 = 0 1 0

Now what? I think I need to understand what's going on, not just what I need to do...