Prove the following statements and illustrate them with examples of your own choice. Here, λ1, ∙ ∙ ∙ λn are the (not necessarily distinct) Eigen values of a given n x n matrix A = [ajk].
(a) Trace the sum of the main diagonal entries is called the trace of A. It equals the sum of the Eigen values.
(b) “Spectral shift”. A – k1 has the Eigen values λ1 – k, ∙∙∙ λn – k and the same Eigen vectors as A.
(c) Scalar multiples, powers. kA has the Eigen values k λ1, ∙∙∙ kλn. Am (m = 1, 2, ∙∙∙) has the Eigen values λ1m, ∙∙∙ λnm. The Eigen vectors are those of A. (d) Spectral mapping theorem, the "polynomial matrix"
Has the Eigen values where j = 1, ∙∙∙ nm and the same Eigen vectors as A.
(e) Peron’s theorem, Show that a Leslie matrix L with positive l12, l13, l21, l32, has a positive Eigen value. (This is a special case of the famous Perron-Frobenius theorem in Sec. 20.7, which is difficult to prove in its general form.).
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