Is x in the nullspace of A an eigenvector of A?

  • Thread starter Thread starter sana2476
  • Start date Start date
  • Tags Tags
    Eigenvalue Proof
sana2476
Messages
33
Reaction score
0
Let x not equal to zero be a vector in the nullspace of A. Then x is an eigenvector of A.


I'm not sure how to start this proof
 
Physics news on Phys.org
If x is a non-zero vector in the null space of A, then you know that A is singular, and you also know that \lambda = 0 is an eigenvalue of A since A is invertible if and only if zero is not an eigenvalue of A. That should start you off.
 
Saying that x is in the null space of A means that Ax= 0= 0x.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

How to find one corresponding eigenvector?
Replies
8
Views
2K
Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.
Replies
5
Views
2K
Direct Proof that every zero of p(T) is an eigenvalue of T
Replies
24
Views
3K
Operator T, ##T^2=I##, -1 not an eigenvalue of T, prove ##T=I##.
Replies
2
Views
1K
Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1
Replies
8
Views
2K
Prove that ##\lambda## or ##-\lambda## is an eigenvalue for ##T##.
Replies
11
Views
2K
Question regarding eigenvectors
Replies
7
Views
2K
Linear homogenous system with repeated eigenvalues
Replies
2
Views
1K
Find eigenvalues & eigenvectors
Replies
3
Views
2K
Check of a problem about nullspace
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top