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Let λ be an eigenvalue of the n × n matrix A and x = 0 be an associated eigenvector.
a. Show that λ is also an eigenvalue of At.
b. Show that for any integer k ≥ 1, λk is an eigenvalue of Ak with eigenvector x.
c. Show that if A−1 exists, then 1/λ is an eigenvalue of A−1 with eigenvector x.
d. Generalize parts (b) and (c) to (A−1)k for integers k ≥ 2.
e. Given the polynomial q(x) = q0 + q1x + · · · + qkxk , define q(A) to be the matrix q(A) = q0I + q1A+· · ·+qkAk . Show that q(λ) is an eigenvalue of q(A) with eigenvector x.
f. Let α ≠ λ be given. Show that if A − αI is nonsingular, then 1/(λ − α) is an eigenvalue of (A − αI)−1 with eigenvector x. |
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