Matrix Eigen Vector Question

  • #1

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...

HtQcr.png


Homework Equations





The Attempt at a Solution



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

Answers and Replies

  • #2
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6,644

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).

HtQcr.png


Homework Equations





The Attempt at a Solution



Part A: All eigenvectors are valid. Eigenvalues are 1, 1/2 and 1/3.
 
  • #3
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...
 
  • #4
34,900
6,644
The ei 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.
 
  • #5
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 .
 
  • #6
34,900
6,644
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 ki's are eigenvectors, you can calculate A ki, and hopefully you have learned something about the eigenvalues of An.
 
  • #7
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...
 

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