## Acalytica

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Let $L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the assertions that are correct.
a) $a \lambda$ is an eigenvalue of $a L$ for any scalar $a$.
b) $\lambda^{2}$ is an eigenvalue of $L^{2}$.
c) $\lambda^{2}+a \lambda+b$ is an eigenvalue of $L^{2}+a L+b I_{n}$ for all real scalars $a$ and $b$.
d) If $\lambda=a+i b$, with $a, b \neq 0$ some real numbers, is an eigenvalue of $L$, then $\bar{\lambda}=a-i b$ is also an eigenvalue of $L$.
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The correct assertions are:

a) $a \lambda$ is an eigenvalue of $a L$ for any scalar $a$.

d) If $\lambda=a+i b$, with $a, b \equiv 0$ some real numbers, is an eigenvalue of $L$, then $\bar{\lambda}=a-i b$ is also an eigenvalue of $L$.

Explanation:

a) If $v$ is an eigenvector of $L$ corresponding to $\lambda$, then $a v$ is an eigenvector of $a L$ corresponding to $a$ $\lambda$.

d) If $\lambda$ is an eigenvalue of $L$ with eigenvector $v$, then $\bar{\lambda}$ is an eigenvalue of $L$ with eigenvector $\bar{v}$.

by Diamond (78,343 points)

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