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Exercise 24 V is a real or complex space. Show that. 1. If the linear map $A: V \rightarrow V$ satisfies $A^2 = A$ and if $\lambda$ is an eigenvalue of $A$, then $\lambda = 0$ or $\lambda = 1$. Give a non-trivial example. 2. If the linear map $A: V \rightarrow V$ satisfies $A^2 = I$ and if $\lambda$ is an eigenvalue of $A$, then $\lambda = 1$ or $\lambda = -1$. Give a non-trivial example. 3. If the linear map $A: V \rightarrow V$ has an inverse and $\lambda$ is an eigenvalue of $A$ then: (a) $\lambda \neq 0$ (b) $1/\lambda$ is an eigenvalue of $A^{-1}$ (c) $E_{1/\lambda}(A^{-1}) = E_\lambda(A)$.

          Exercise 24
V is a real or complex space. Show that.
1. If the linear map $A: V \rightarrow V$ satisfies $A^2 = A$ and if $\lambda$ is an eigenvalue of $A$, then $\lambda = 0$ or $\lambda = 1$.
Give a non-trivial example.
2. If the linear map $A: V \rightarrow V$ satisfies $A^2 = I$ and if $\lambda$ is an eigenvalue of $A$, then $\lambda = 1$ or $\lambda = -1$. Give a non-trivial example.
3. If the linear map $A: V \rightarrow V$ has an inverse and $\lambda$ is an eigenvalue of $A$ then:
(a) $\lambda \neq 0$
(b) $1/\lambda$ is an eigenvalue of $A^{-1}$
(c) $E_{1/\lambda}(A^{-1}) = E_\lambda(A)$.

Exercise 24
V is a real or complex space. Show that.
1. If the linear map A: V → V satisfies A^2 = A and if λ is an eigenvalue of A, then λ = 0 or λ = 1.
Give a non-trivial example.
2. If the linear map A: V → V satisfies A^2 = I and if λ is an eigenvalue of A, then λ = 1 or λ = -1. Give a non-trivial example.
3. If the linear map A: V → V has an inverse and λ is an eigenvalue of A then:
(a) λ≠ 0
(b) 1/λ is an eigenvalue of A^-1
(c) E1/λ(A^-1) = Eλ(A).

Added by Maurice P.

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