Eigenvalues of $Q=I+2P$

by user481975   Last Updated May 16, 2019 06:20 AM

It is a question of PHD entrance.In my opinion (a) is correct.Pleease help me. I have tried to do it evaluated option (a). I think it is correct.Can not get the other options.Please help me.

Answers 1

Here are some hints to help you get started.

Part c) Suppose $v$ is an eigenvector of $P$ with eigenvalue $\lambda$, i.e. $Pv = \lambda v$. Then $Qv = (I+2P)v = v+2Pv = v+2\lambda v = (1+2\lambda)v$, i.e. $v$ is an eigenvector of $Q = I+2P$ with eigenvalue $1+2\lambda$. So if $-1$ and $+1$ are the possible eigenvalues of $P$, what are the possible eigenvalues of $Q$?

Part b) A matrix is invertible if and only if all the eigenvalues are non-zero. Use your answer to part c to answer this.

Part d) The determinant of a matrix is the product of its eigenvalues (counting multiplicity). So $\det P$ is the product of a bunch of $-1$'s and $+1$'s. Hence, $\det P > 0$ if $P$ has an even number of eigenvalues that are $-1$ and $\det P < 0$ if $P$ has an odd number of eigenvalues that are $-1$. Can you make a similar statement about $Q$? How are the eigenvalues of $P$ and $Q$ related?

Part a) If $Q$ is invertible (see part b), then since $\{a_1,\ldots,a_n\}$ is a basis, so is $\{Qa_1,\ldots,Qa_n\}$. But $Qa_i = (I+2P)a_i = \cdots$.

May 16, 2019 06:17 AM

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