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Problem 1 (Diagonalisation) Let A == 5 -4 2 4 -2 2 0 -3 2 € R3x3 a) Show that λ = 1 is an eigenvalue of A with geometric multiplicity 2, and determine a basis for the corresponding eigenspace. 2 b) Show that v := 2 is an eigenvector of A, and calculate the associated eigenvalue. c) Show that A is diagonalisable and find an invertible matrix X E R3x3 and a diagonal matrix D such that X-¹AX = D.

Problem 1 (Diagonalisation) Let A == 5 -4 2 4 -2 2 0 -3 2 € R3x3 a) Show that λ = 1 is an eigenvalue of A with geometric multiplicity 2, and determine a basis for the corresponding eigenspace. 2 b) Show that v := 2 is an eigenvector of A, and calculate the associated eigenvalue. c) Show that A is diagonalisable and find an invertible matrix X E R3x3 and a diagonal matrix D such that X-¹AX = D.

Advanced Engineering Mathematics
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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Problem 1 (Diagonalisation) Let 5 -4 2 A== -- (1:0) = -2 2 -3 2 ER³X3 a) Show that \ and determine a basis for the corresponding eigenspace. - 1 is an eigenvalue of A with geometric multiplicity 2, b) Show that v := (3) 2 is an eigenvector of A, and calculate the associated eigenvalue. c) Show that A is diagonalisable and find an invertible matrix X E R³×3 and a diagonal matrix D such that X-¹AX = D. d) Find a 3 x 3-matrix B such that B2 = A.
Transcribed Image Text:Problem 1 (Diagonalisation) Let 5 -4 2 A== -- (1:0) = -2 2 -3 2 ER³X3 a) Show that \ and determine a basis for the corresponding eigenspace. - 1 is an eigenvalue of A with geometric multiplicity 2, b) Show that v := (3) 2 is an eigenvector of A, and calculate the associated eigenvalue. c) Show that A is diagonalisable and find an invertible matrix X E R³×3 and a diagonal matrix D such that X-¹AX = D. d) Find a 3 x 3-matrix B such that B2 = A.
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