Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given a Linear Operator T and a vector or linear combination of vectors from within a single eigenspace of T, will those vectors always be projected back into that eigenspace when given as input to T? I tend to think yes since that kind of seems to be the definition of an eigenvector.

share|improve this question
add comment

2 Answers 2

up vote 1 down vote accepted

Supposing that $\vec v_1,...,\vec v_n$ are vectors in the eigenspace of $T$ corresponding to the eigenvalue $\lambda$, and if $a_1,...,a_n$ are constants, then we have $$T(a_1\vec v_1+...+a_n\vec v_n)=a_1T(\vec v_1)+...+a_nT(\vec v_n)=a_1\lambda\vec v_1+...+a_n\lambda\vec v_n=\lambda(a_1\vec v_1+...+a_n\vec v_n),$$ so you're correct.

share|improve this answer
add comment

Yes. Let $v_i$ be an eigenvector to the eigenvalue $\lambda_i$. Let $x=\sum_{i=1}^n \mu_i v_i$, then $Tx = \sum_{i=1}^n \lambda_i \mu_i v_i$.

To prove your case, let $\lambda_i = \lambda_j =: \lambda$ for $i,j$. Then $Tx = \lambda x$ by the above calculation.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.