Question 4.E.8.8: Let λ be a scalar such that (C − λI)n×n is singular.(a) If B......

Matrix Analysis and Applied Linear Algebra [1113429]

Let λ be a scalar such that $(C − λI)_{n×n}$ is singular.

(a) If $B \simeq C$ , prove that (B − λI) is also singular.

(b) Prove that $(B − λ_{i}I)$ is singular whenever $B_{n×n}$ is similar to

D = \begin{pmatrix}λ_1& 0 &· · ·& 0\\0 &λ_2& · · ·& 0\\\vdots&\vdots&\ddots&\vdots\\0 &0 &· · ·& λ_n\end{pmatrix}.

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(a) $B = Q^{−1}CQ \Longrightarrow (B − λI) = Q^{−1}CQ − λQ^{−1}Q = Q^{−1}(C − λI)Q.$ The result follows because multiplication by nonsingular matrices does not change rank.

(b) $B = P^{−1}DP \Longrightarrow B − λ_{i}I = P^{−1}(D − λ_{i}I)P and (D − λ_{i}I)$ is singular for each $λ_{i}$ . Now use part (a).

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