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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![8. Let \( A \in \mathbb{C}^{n \times n} \) and let \( \mathcal{X} \subseteq \mathbb{C}^n \) be a subspace of \( \mathbb{C}^n \). Recall that \( \mathcal{X} \) is said to be an \( A \)-invariant subspace if \( A(\mathcal{X}) := \{ Ax : x \in \mathcal{X} \} \subseteq \mathcal{X} \).
The rest of this problem concerns the matrix \( A - \lambda I \), where \( \lambda \in \mathbb{C} \) and \( I = I_n \).
(a) Prove that \( (A - \lambda I)^\ell \cdot A = A \cdot (A - \lambda I)^\ell \) for all \( \ell = 0, 1, 2, \ldots \).
(b) Assume that \( \lambda \) is a scalar such that \( (A - \lambda I) \) is singular, and recall from the first problem that there exists the smallest positive integer \( k \) such that
\[
N((A - \lambda I)^k) = N((A - \lambda I)^{k+1}) \quad \text{and} \quad R((A - \lambda I)^k) = R((A - \lambda I)^{k+1}).
\]
Prove that \( N((A - \lambda I)^k) \) and \( R((A - \lambda I)^k) \) are \( A \)-invariant subspaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2F00ec1d06-392e-4112-930b-8aaffe989c6a%2Fgg6p3hr_processed.png&w=3840&q=75)
Transcribed Image Text:8. Let \( A \in \mathbb{C}^{n \times n} \) and let \( \mathcal{X} \subseteq \mathbb{C}^n \) be a subspace of \( \mathbb{C}^n \). Recall that \( \mathcal{X} \) is said to be an \( A \)-invariant subspace if \( A(\mathcal{X}) := \{ Ax : x \in \mathcal{X} \} \subseteq \mathcal{X} \).
The rest of this problem concerns the matrix \( A - \lambda I \), where \( \lambda \in \mathbb{C} \) and \( I = I_n \).
(a) Prove that \( (A - \lambda I)^\ell \cdot A = A \cdot (A - \lambda I)^\ell \) for all \( \ell = 0, 1, 2, \ldots \).
(b) Assume that \( \lambda \) is a scalar such that \( (A - \lambda I) \) is singular, and recall from the first problem that there exists the smallest positive integer \( k \) such that
\[
N((A - \lambda I)^k) = N((A - \lambda I)^{k+1}) \quad \text{and} \quad R((A - \lambda I)^k) = R((A - \lambda I)^{k+1}).
\]
Prove that \( N((A - \lambda I)^k) \) and \( R((A - \lambda I)^k) \) are \( A \)-invariant subspaces.
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