Task 3: Let F: R4 → R4 be a linear transformation given by F(x) = Ax where A is any (4 x 4) matrix that has rank 2 and satisfies A² = 3A. a) Show that F has exactly two eigenvalues and determine these. (Hint: what applies to A² v if is an eigenvector of F) b) Determine the algebraic and geometric multiplicity of each of the two eigenvalues of F. (Hint: compare the eigenspace to F with the kernel and the image to F. Give an example of a matrix A that meets the conditions of the task. c)
Task 3: Let F: R4 → R4 be a linear transformation given by F(x) = Ax where A is any (4 x 4) matrix that has rank 2 and satisfies A² = 3A. a) Show that F has exactly two eigenvalues and determine these. (Hint: what applies to A² v if is an eigenvector of F) b) Determine the algebraic and geometric multiplicity of each of the two eigenvalues of F. (Hint: compare the eigenspace to F with the kernel and the image to F. Give an example of a matrix A that meets the conditions of the task. c)
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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Transcribed Image Text:Task 3: Let F: R4 → R4 be a linear transformation given by F(x) = Ax where A is any (4 x 4) matrix
that has rank 2 and satisfies A² = 3A.
a) Show that F has exactly two eigenvalues and determine these. (Hint: what applies to A² v if
is an eigenvector of F)
b)
Determine the algebraic and geometric multiplicity of each of the two eigenvalues of F.
(Hint: compare the eigenspace to F with the kernel and the image to F.
Give an example of a matrix A that meets the conditions of the task.
c)
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