8. Let A € Cnxn and let X Cn be a subspace of Cn. Recall that is said to be an A-invariant subspace ifA(X) = {Ax : XE X} CX.The rest of the this problem concerns the matrix A - XI, where λ = C and I = In.(a) Prove that (A — XI)² · A = · A · (A — XI) for all l = 0, 1, 2, ....(b) Assume that A is a scalar such that (A - XI) is singular, and recall from the first problem that there existsthe smallest positive integer k such thatN((A–XI)*)N((A–\I)*+1)Prove that N ((A – XI)k) and R((A – XI)k) . are A-invariant subspaces.=and R((A — XI)K) R((A — XI)k+¹).

As per the question we are given a matrix A ∈ Cn×n and we have to show that : (A - λI)l • A = A •…

Let A E Cnxn and let XC Cn be a subspace of Cn. Recall that is said to be an A-invariant subspace if A(X) := {Ax : x = X} C X. The rest of the this problem concerns the matrix A – XI, where λ = C and I = In. (a) Prove that (A — XI)¤ · A = A · (A – XI)e for all l = 0, 1, 2, . . .. - (b) Assume that A is a scalar such that (A – XI) is singular, and recall from the first problem that there exists the smallest positive integer k such that N((A–XI)*) = N((A–\I)*+1) and R((A - XI)K) Prove that N ((A - \I)*) and R((A — \I)*) are A-invariant subspaces. = R((A - XI)K+¹).

Let A E Cnxn and let XC Cn be a subspace of Cn. Recall that is said to be an A-invariant subspace if A(X) := {Ax : x = X} C X. The rest of the this problem concerns the matrix A – XI, where λ = C and I = In. (a) Prove that (A — XI)¤ · A = A · (A – XI)e for all l = 0, 1, 2, . . .. - (b) Assume that A is a scalar such that (A – XI) is singular, and recall from the first problem that there exists the smallest positive integer k such that N((A–XI)) = N((A–\I)+1) and R((A - XI)K) Prove that N ((A - \I)) and R((A — \I)) are A-invariant subspaces. = R((A - XI)K+¹).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 30E

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