Eigenvalue of matrix proof by induction

TanWu
Messages
17
Reaction score
5
Homework Statement
Show that if ##\alpha## is an eigenvalue of matrix ##B## then ##\alpha^{n}## is an eigenvalue of matrix ##B^{n}## for positive integer ##n##
Relevant Equations
##\alpha## is eigenvalue
##B\vec x = \alpha \vec x##
We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds.

Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x##

Now consider ##n = m## case,

##B^m\vec x = B(B^{m - 1} \alpha) = B^m \alpha = \alpha^m \vec x##

Now consider ##n = m + 1## case,
##B^{m + 1}\alpha = B(B^m \alpha) = B(\alpha^m \vec x) = \alpha^m(B\vec x) = \alpha^k(\alpha \vec x) = \alpha^{m + 1}\vec x##

Is that correct use of induction?
gratitude expressed to those who help.
 
Last edited by a moderator:
Physics news on Phys.org
You can formulate more precisely. Instead of saying "consider ##n=m## case", we can say "assume that ## B^nx = \alpha ^nx ## for some ##n\geqslant 2##". Then for the case ##n+1## we have the equalities
<br /> B^{n+1}x = BB^nx = B\alpha ^nx = \alpha ^nBx = \alpha ^n\alpha x = \alpha ^{n+1}x.<br />
Right now you haven't made it explicit what your induction assumption is. Anyone familiar with the technique is able to fill in the gaps.
 
TanWu said:
Homework Statement: Show that if ##\alpha## is an eigenvalue of matrix ##B## then ##\alpha^{n}## is an eigenvalue of matrix ##B^{n}## for positive integer ##n##
Relevant Equations: $\alpha$ is eigenvalue
##B\vec x = \alpha \vec x##

We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds.

Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x##
You shouldn't need to show it for ##n=2##. After the base case, state the induction hypothesis:
$$B^m\vec x = \alpha^m \vec x$$
is true for some integer ##m>1##.

Apply the induction hypothesis to show its true for ##m+1##, as you have done so. Then you can conclude it's true for all integers.
 
It is not wrong to check the claim up to ##m## manually. Sometimes that's even helpful. The induction hypothesis then is that the claim is true
  1. for some ##n\geqslant m## (weak induction);
  2. for all ##k\leqslant n##, where ##n\geqslant m## (strong induction).
In both cases the task is to prove the claim is true for ##n+1##. It doesn't matter which form of induction we use, they are equivalent.
 
Last edited:
Sounds good, though I feel it's more concise/elegant to use weak induction here. But just to clarify, the claim being true for ##n+1##, or proving the claim is true for ##n+1##, is not any part of the induction hypothesis. Rather, it's the consequence that would follow if the induction hypothesis is true.
 
Fair play on your last point. As for the first one, optimisation can be done later. First and foremost, it is important the argument is correct.
 
@PeroK I have a nagging feeling that my statement in #5 about the induction hypothesis is not correct. If you have a moment, can you please point out what's wrong about it? Thank you.
 
docnet said:
But just to clarify, the claim being true for n+1, or proving the claim is true for n+1, is not any part of the induction hypothesis.
Right. The induction hypothesis in this case is assuming that ##B^m\vec x = a^m\vec x## is true. Using this hypothesis, you show that it follows that ##B^{m+1}\vec x = a^{m+1}\vec x## must also be true. Of course you need to also establish that some base case is true.
 
  • Like
Likes nuuskur, PeroK and docnet
Back
Top