- PDF - text ### Text Suppose 𝑇 ∈ ℒ (𝑉). Prove that 9 is an eigenvalue of $𝑇^2$ if and only if 3 or −3 is an eigenvalue of 𝑇. SOLUTION First suppose 3 is an eigenvalue of 𝑇. Then there exists 𝑣 ∈ 𝑉 with 𝑣 ≠ 0 such that 𝑇𝑣 = 3𝑣 . Applying 𝑇 to both sides, we get $𝑇^2$𝑣 = 3𝑇𝑣 = 9𝑣. Thus 9 is an eigenvalue of $𝑇^2$ Similarly, if −3 is an eigenvalue of 𝑇 , then 9 is an eigenvalue of $𝑇^2$ . Now suppose 9 is an eigenvalue of $𝑇^2$. Thus $𝑇^2$− 9𝐼 is not injective. Because $𝑇^2$ − 9𝐼 = (𝑇 + 3𝐼)(𝑇 − 3𝐼) , this implies that 𝑇 + 3𝐼 is not injective or 𝑇 − 3𝑖 is not injective. Thus −3 is an eigenvalue of 𝑇 or 3 is an eigenvalue of 𝑇. Edward Frenkel ---