How can I show that y is an eigenvalue of B?

  • Thread starter gimpy
  • Start date
In summary, The conversation is about solving two problems involving matrices and eigenvectors. The first problem asks to show that a given eigenvector is also an eigenvector of a different matrix, and to find the corresponding eigenvalue. The second problem asks to show that two matrices are similar and involves using the invertibility of one of the matrices. The conversation includes some confusion and typos, but ultimately leads to finding the correct solutions by rearranging the equations.
  • #1
gimpy
28
0
Ok well i have two questions.

1) If B = P^-1AP and let X be an eigenvector of A corresponding to the eigenvalue y. Show that y is an eigenvalue of B and find a corresponding eigenvector.

This is what i did.
AX=yI and since B = P^-1Ap -> A = PBP^-1
so (PBP^-1)X=yI
Now this is the part where i get lost. Am i on the right track?

2) If A and B are nxn matrices, A is invertable, show that BA is similar to AB.

So BA = P^-1ABP because BA is similar to AB. But I am kinda lost now. I'm sure i have to do something with the fact that A is invertable. Umm... A^-1...
 
Physics news on Phys.org
  • #2
1)

AX=yI

Here's your mistake; what you should have is Ax = yx.

Recall that the definition is:

[itex]\vec{v}[/itex] is an eigenvector of [itex]A[/itex] corresponding to eigenvalue [itex]\lambda[/itex] iff [itex]A\vec{v} = \lambda \vec{v}[/itex].


2)

Ok, you know to be similar, you need

[tex]BA = P^{-1}ABP[/tex]

What's the simplest guess as to what P should be to make this equation hold?
 
Last edited:
  • #3
Originally posted by Hurkyl
1)



Here's your mistake; what you should have is Ax = yx.

Recall that the definition is:

[itex]\vec{v}[/itex] is an eigenvector of [itex]A[/itex] corresponding to eigenvalue [itex]\lambda[/itex] iff [itex]A\vec{v} = \lambda \vec{v}[/itex].


2)

Ok, you know to be similar, you need

[tex]BA = P^{-1}ABP[/tex]

What's the simplest guess as to what P should be to make this equation hold?

lol

i meant [itex]A\vec{v} = \lambda \vec{v}[/itex]. Stupid typo by me. But i still don't get it [b(]

And for the other one, I am lost as to what [itex]P[/itex] should be to make this equation hold. It says that [tex]A[/tex] and [itex]B[/itex] are nxn matrices. [itex]A[/itex] is invertable. show that [itex]BA[/itex] is similar to [itex]AB[/itex].. umm...
 
  • #4
1)

You know that you need to find something of the form [itex]B\vec{w}=\lambda\vec{w}[/itex]...

but you have something of the form [itex]P^{-1}BP\vec{v} = y \vec{v}[/itex]...

The first thing I notice is that the LHS must be the matrix [itex]B[/itex] times some vector... so can you rewrite what you do have in such a way that the LHS is [itex]B[/itex] times some vector?


2)

You need to find a [itex]P[/itex] such that:

[tex]BA = P^{-1}ABP[/tex]


The first thing I notice is that I need to have an [itex]A[/itex] on the right of [itex]B[/itex]... you have a [itex]B[/itex] on the right hand side, can you select a value for [itex]P[/itex] so that we have an [itex]A[/itex] on the right of the [itex]B[/itex]?
 

1. How would I show this experiment to others?

To show your experiment, you could create a presentation using visual aids such as charts, graphs, and images. You could also create a video demonstration or a written report with detailed descriptions and data analysis. Additionally, you could invite others to observe the experiment in person.

2. How would I visually represent my findings?

To visually represent your findings, you could use graphs, charts, and images to display your data and results. You could also use diagrams or illustrations to explain your methods and observations.

3. How would I present my research to a non-scientific audience?

To present your research to a non-scientific audience, you could use simple and clear language to explain your methods, results, and significance of your findings. You could also use visual aids and real-life examples to make your research more relatable and understandable.

4. How would I make my experiment reproducible?

To make your experiment reproducible, you could provide detailed descriptions of your methods, materials, and procedures used. You could also make your data and results available for others to analyze and replicate.

5. How do I ensure the accuracy of my experiment?

To ensure the accuracy of your experiment, you could conduct multiple trials and use control groups to compare your results. You could also have your experiment reviewed by other scientists or peers for feedback and suggestions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
413
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
40
Views
3K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
15
Views
2K
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
960
  • Linear and Abstract Algebra
Replies
2
Views
494
Replies
7
Views
782
  • Advanced Physics Homework Help
Replies
17
Views
1K
Back
Top